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INTRODUCTION 


TO 


INFINITE    SERIES 


OSGOOD 


SCENCE&ENGINEEBING 
UBBABV 

AUG  2  7  1997 

Ews  coaecTioN 

UCLA 


s- 


n, 


sJuZ-/n-<\c 


%au 


/^  I  9 


INTRODUCTION 


TO 


INFINITE    SERIES 


BY 


WILLIAM   F.   OSGOOD,  Ph.D.,  LL.D. 

PROFESSOR    OF    MATHEMATICS    IN    HARVARD   UNIVERSITY 


XHIRID    EDITION 


SCIENCE  &  ENGINEERING 
LIBRARY 

AUG  2  7  1997 

EMS  COLLECTION 
UCLA 


CAMBRIDGE 

publisbcCt  bY>  l3arvar^  *Cluivev5it\? 

1910 


1  •  •( 


Copyright,  1897,  by 
Harvard  University. 


First  edition,  April,  1897. 

Reprinted  with  corrections,  September,  1902. 

Reprinted,  January,  1910. 


PREFACE. 


TN  an  introductory  course  on  the  Differential  and  Integral  Calculus 
the  subject  of  Infinite  Series  forms  an  important  topic.  The 
presentation  of  this  subject  should  have  in  view  first  to  make  tlie 
beginner  acquainted  with  the  nature  and  use  of  infinite  series  and 
secondly  to  introduce  him  to  thi'  theory  of  these  series  in  such  a  way 
that  he  sees  at  each  step  precisely  what  the  question  at  issue  is  and 
never  enters  on  the  proof  of  a  theorem  till  he  feels  that  the  theorem 
actually  requires  proof.  Aids  to  tlu'  attainment  of  these  ends  are : 
(a)  a  variety  of  illustrations,  taken  from  the  cases  that  actually  arise 
in  practice,  of  the  application  of  series  to  computation  both  in  pure 
and  applied  mathematics ;  (b)  a  full  and  careful  exposition  of  the 
meaning  and  scope  of  the  more  ditlicult  theorems ;  (c)  the  use  of 
diagrams  and  graphical  illustrations  in  the  proofs. 

The  pamphlet  that  follows  is  designed  to  give  a  presentation  of 
the  kind  here  indicated.  The  references  are  to  Byerly's  Differential 
Calculus,  Integral  Calculus,  and  Problems  in  Differential  Calculus; 
and  to  B.  O.  Peirce's  Short  Table  of  Integrals;  all  published  by 
Ginn  &  Co.,   Boston. 

WM.  F.   OSGOOD. 

Cambridge,  April  1897. 


(/S2^ 


'■y 


introductio:n". 


1.   Examj^le.  —  Consider  the  successive  values  of  the  variable 

s,.  r=  1  4-r+r2+ _^,.n-: 

for  w  =  1,  2,  3, Let  r  have  the  value  ^.     Then 

s,=  l  =i 

s,  =  1  +  -^  =1^ 

S3    =    1    +    J   +    i  =1^ 

■  ••••• 

If  the  values  be  represented  by  points  on  a  line,  it  is  easy  to  see  the 
0  S,  =  I  S^      S,  3^2. 

— H 1 1 — rill 


Fiu.  1. 

law  by  which  any  s„  can  be  obtained  from  its  predecessor,  s„_i, 
namely  :  .s„  lies  half  way  between  s„_i  and  2. 

Hence  it  appears  that  wlien  )i  increases  without  limit, 

Lim  .s„  =  2. 

The  same  result  could  have  been  obtained  arithmetically  from  the 
formula  for  the  sum  s„  of  the  first  n  terms  of  the  geometric  series 

a-\-  ar-\-  ar^  -\- +  ar"-\ 

a(l— r") 

'"  -  1   —  7- 

Here  a  =;  1,  r  =  i, 

1-i 

2"        ,  1 

=  2  — 


"  1  9'i  —  1 

^  ~2 

When  /(   increases  without  limit,  ^^i^^i  approaches  0  as  its  limit, 
and  hence  as  before  Lim  s„  ==  2. 


2  INTRODUCTION.  §  2. 

2.    Definition  of  an  Infinite  Series.     Let  ?<o?  ''^i?  "^a? be 

any  set  of  values,  positive  or  negative  or  both,  and  form  the  series 

^0   +    ^<l    +    ^2   + (1) 

Denote  the  sum  of  the  first  n  terms  by  s„ : 

s„  =  Uo  +  Ui-\- +  ti„_i. 

Allow  n  to  increase  without  limit.  Then  either  a)  s„  will  approach 
a  limit  U: 

Lim  s„  =  U; 

W=  CO 

or  b)  s„  approaches  no  limit.  In  either  case  we  speak  of  (1)  as  an 
Infinite  Series,  because  n  is  allowed  to  increase  without  limit.  In 
case  a)  the  infinite  series  is  said  to  be  convergent  and  to  have  the 
value*  U  ov  converge  towards  the  value  U.  In  case  h)  the  infinite 
series  is  said  to  be  divergent. 

The  geometric  series  above  considered  is  an  example  of  a  con- 
vergent series. 

1+2+3+   , 

1-1+1-  

are  examples  of  divergent  series.     Only  convergent  series  are  of  use 
in  practice. 
The  notation 

r/.Q  +  ?'i  + ad  inf.  (or  to  infinity) 

is  often  used  for  the  limit  U",  or  simply 

U  =  Uq-\-  u^-\- 

*  U\s  often  called  the  sum  of  the  series.  But  the  student  must  not  forget 
that  U  is  7iot  a  sum,  but  is  the  limit  of  a  sum.  Similarly  the  expression  "the  sum 
of  an  infinite  number  of  terms"  means  the  limit  of  the  sum  of  n  of  these  terms, 
as  n  increases  without  limit. 


I.  COXYERGENCE. 


a)    SERIES,    ALL    OF   WHOSE   TERMS    ARE    POSITIVE. 

3.    Example.     Let  it  be  required  to  test  the  convergence  of  the 
series 

'  +  '  +  F2  +  riT3+ +^+ (^) 

where  ?i !    means   1-2-3 n  and  is  read  ^^ factorial  n". 

Discarding  for  the  moment  the  first  term,  compare  the  sum  of  the 
next  n  terms 

_1  1  1 

°'"  ~  ^    •"  F2  "^  1  •  2  •  3  ~^ ~^  i-2-3 n 

with  the  corresponding  sum 

1  1  1 


2    '    2-2    '  '    2 -2 -2 


n  —  1  factors 


=  2-^,<2     (Cf.  §1), 


Each  term  of  a,,  after  the  first  two  is  less  than  the  corresponding 
term  in  S,,^  and  hence  the  sum 

or,  inserting  the  discarded  term  and  denoting  the  sum  of  the  first  n 
terms  of  the  given  series  by  s„, 

s„  +  i=l-hl  +  p^  +  YT2T3-+ +  1-2-3 n<^' 

no  matter  how  large  n  be  taken.  That  is  to  say,  s„  is  a  variable 
that  always  increases  as  n  increases,  but  that  never  attains  so  large 
a  value  as  3.  To  make  these  relations  clear  to  the  eye,  plot  the 
successive  values  of  s„  as  points  on  a  line. 


CONVERGENCE. 

Sl=    1 

=  1 

S2  =  1  +  1 

—  2 

§§3,4. 


53  =    1    +    1    +   iy 

54  =    1    +    t    +   ^  +   — 

,1.1,1 

S5  =1  +  1  +  27+31  +  47 


=  2.5 
—  2.667 
r=  2.708 


s«—  1  +  1  +  — +  —  +  —  +  —  =2.717 

s,  —  1  +  1  +  —  +  —  +  —+  —  +  -^  =  2.718 


2  !    '    3 !    '    4 !    '    5 !    '    6 

1    .    .i  +  l  +  J_  +  i 
2!^3!^4!~5!'6! 


o    —  i_|_i4_JL_|_.l.-i_JL_|_J__|__L_(__L—  2.718 


o  s,=  i  s^=a      s^s-    3 


4 H \ 4-th-+ 


FiQ.  2. 


When  n  increases  by  1 ,  the  point  representing  s„  +  ^  always  moves 
to  the  right,  bnt  never  advances  so  far  as  the  point  3.  Hence  there 
must  he  some  point  e,  either-  coinciding  with  3  or  lying  to  the  left  of  3 
(i.e.  e  <C  3),  which  s„  apj^roaches  as  its  limit,  but  never  reaches.     To 

judge  from  the  vakies  computed  for  s^,  Sj,  •  •  •  Sg,  the  value  of  e  to 
four  significant  figures  is  2.71<S,  a  fact  that  will  be  established 
later. 

4.  Fundamental  Principle.  The  reasoning  by  which  we  have 
here  inferred  the  existence  of  a  limit  e,  althongh  we  do  not  as  yet 
know  how  to  compute  the  numerical  value  of  that  limit,  is  of  prime 
importance  for  the  work  that  follows.  Let  us  state  it  clearly  in 
general  form. 

//'.s„  is  a  variable  that  1)  always  increases  ivhen  n  increases: 

s,,  >  s„ ,  w'  >  ?t ; 

but  that  2)  always  remains  less  than  some  definite  fixed  number,  A: 

s„  <  A 

for  all  values  ofn,  then  s„  apjrroaches  a  limit,  U: 

Lim  .s„  =1  U. 
n  =  00 


§§  4,  5.  CONVEUGEXCE.  5 

This  limit,  U,  is  not  greater  than  A : 

U<A. 

S.  Sa  Sa       S.     U  A 

i \ 1        I    Mil \ 

Fio.  3. 

The  value  A  may  be  the  limit  itself  or  any  value  greater  than  the 
limit. 

Exercise.  State  the  Principle  for  a  variable  that  is  always  de- 
creasing, but  is  always  greater  than  a  certain  fixed  quantity,  and 
draw  the  coiTesponding  figure. 

5.  First  Test  for  Convergence.  Direct  Comparison.  On  the  prin- 
ciple of  the  preceding  paragraph  is  based  the  following  test  for  the 
convergence  of  an  infinite  series. 

Let  Kq  +  "i  +  ^2  + (a) 

be  a  series  of  positive  terms,  the  convergence  of  which  it  is  desired  to 
test.     If  a  series  of  positive  terms  already  known  to  he  convergent 

<'0   +    ^'l    +    «2   + (/?) 

can  he  found  lohose  terms  are  never  less  than  the  corresponding  terms 
in  the  series  to  he  tested  (a),  then  (a)  is  a  convergent  series,  and  its  value 
does  not  exceed  that  of  the  series  (fi). 
For  let 

S„=  a^-j-  a,-\- -f  «,._!, 

Lim  S„  =  A. 
n  =  00 

Then  since  S„  <^  A  and  s„  ^  S„ , 

it  follows  that  s„  <^  A 

and  hence  by  §4  s„  approaches  a  limit  and  tliis  limit  is  not  greater 
than  A. 

Remark.  It  is  frequently  convenient  in  studying  the  convergence 
of  a  series  to  discard  a  few  terms  at  the  beginning  (»i,  say,  when  m 
is  Vi  fixed  number)  and  to  consider  the  new  series  thus  arising.  That 
the  convergence  of  this  series  is  necessary  and  sufficient  for  the 
convergence  of  the  original  series  is  evident,  since 

S«   =    ("O   -f +    "„.-l)    +    ("«    + -f-    «n-l) 


6  CONVERGENCE.  §§  5,  6. 

u  is  constant  and  hence  s„  will  converge  toward  a  limit  if  s„_„  does, 
and  conversely. 

Examples.     Prove  the  following  series  convergent. 

(1)       1+^4-^  +  ^4+ 

(2)  r  +  r"  +  r«  +  r^'  -f......  0<r<l 

1,1,1. 

^'^         31+5!  +  ^+ 

1,1,1, 
W  1^2  +  2^3  +  3^+ 

Solution.     Write  s„  in  the  form 

1\     ,     /I         1\    ,  ,     /I  1      \ 


^"-^1-2      +      2-3      + +U 


-1 ^. 

~  n  +  1  ' 

then 

Lim  s„  =  1. 


W  =  00 


1-2  ^  3-4  ^  5-6  ^ 


1  +  1  +  1 
22  ^  32  ^  4-^ 


^      I       02      I       02      1       /12      1 


1+^  +   ^+ '  P>^- 


6.    A  New  Test-Series.     It  has  just  been  seen  that  the  series 

1  +  ^  +  ^.  +  ^+ (3) 

converges  when  the  constant  quantity  p  ^  2.  We  will  now  prove 
that  it  also  converges  whenever  p  ^  1.  The  truth  of  the  following 
inequalities  is  at  once  evident : 

-  +  -<-  =  — 

2?    '    3?  ^  2^        2^~  1 

I      Kp     I      fip     I      7/>    ^^   /i/'  AP-1 


4P    I    5?    '    gp    I    7/'  ^  4/-        4, 

i+i+ +i<l  =  _ 

gp    I    9P    I  '    IS''  ^  8"       8 


§§  (),  7.  CONVEKGENCE. 

Hence,  adding  m  of  these  inequalities  togetlier,  we  get 

1,1  I 4.__L__^J__L_L+J_4. 

2*"       6''  (2'"'^^ 1)''       2''""^       4^-1   "^  8''"^ 

+    ' 


Denote  1/2''"^  by  r ;  then,  since  ^>  —  1  ^  0,  r  <^  1  and  the  series 

converges  toward  the   hmit .     Consequently   no  matter  how 

many  terms  of  the  series 

i  +  l  +  i+ 

be  taken,  their  sum  will  always  be  less  than ,  and  this  series  is 

therefore  convergent,  by  the  principle  of  §  4. 

Series  (3)  is  useful  as  a  test-series,  for  many  series  that  could  not 
be  shown  to  be  convergent  by  the  aid  of  the  geometric  series,  can 
be  so  shown  by  reference  to  it.     For  example. 


7.    Divergent  Series.     The  series  (3)  has  been  proved  convergent 
for  every  value  of  j9  ^  1.     Thus  the  series 

1  +  -^  +  ^+^-+ 

2  'V  2        3-^3        A'^  4: 

is  a  convergent  series,  for  j7  :=  1.01.  Now  consider  what  the  nu- 
morioal  values  of  these  roots  in  the  denominators  are  : 

'7  2=1.007,      '7  3  =  1.011,      '7  4=1.014. 

In  fact  '7  100=  1.047  and  '7  1000=  1.071;  that  is,  when  a 
thousand  terms  of  the  series  have  been  taken,  the  denominator  of  the 
last  term  is  multiplied  by  a  number  so  slightly  different  from  1  that 
the  first  significant  figure  of  the  decimal  part  appears  only  in  the 
second  place.  And  wlu'u  one  considers  that  these  same  relations  will 
be  still  more  strongly  marked  when  j)  is  set  equal  to  1.001  or  1.0001, 
one  may  well  ask  whether  the  series  obtained  by  putting  jj  =^  1, 

'  +  l  +  l  +  \+ w 

is  not  also  convergent. 


8  CONVERGENCE.  §§   7,  8. 

This  is  however  not  the  case.     For 

1,1,  J 1       ^        1         1 

^  n  -\-  n  ^      2  n        2  ' 

since  each  of  the  n  terms,  save  the  last,  is  greater  than  1/2  n.  Hence 
we  can  strike  in  in  the  series  anywhere,  add  a  definite  number  of 
terms  together  and  thus  get  a  sum  greater  than  h,  and  we  can  do 
this  as  often  as  we  please.     For  example, 

1    I    1  ^    1 

''  =  '^  3  +  4>2 

''=^^                               5  +  6  +  7  +  8>2 
"  =  ^'  9  +  To  + +  1-6  >  2 


Hence  the  sum  of   the   first  n  terms  increases  without  limit  as  n 
increases  without  limit, 

or  Lim  .S,,  rrr    30 

The  series  (4)  is  called  the  harmonic  series. 

How  is  the  apparently  sudden  change  from  convergence  for  p  ^  1 
in  series  (3)  to  divergence  when  p  =  1  to  be  accounted  for?  The 
explanation  is  simple.  When  p  is  only  slightly  greater  than  1, 
series  (3)  indeed  converges  still,  but  it  converges  towards  a  large 
value,  and  this  value,  which  is  of  coui'se  a  function  of  p,  increases 
without  limit  when  j>,  decreasing,  approaches  1.  When  p  =  1,  no 
limit  exists,  and  the  series  is  divergent. 

8.  Test  for  Divergence.  Exercise.  Establish  the  test  for  diver- 
gence of  a  series  corresponding  to  the  test  of  §5  for  convergence, 
namely  :   Let 

^'o  +  ^'i  + (a) 

he  a  series  of  positive  terms  that  is  to  be  tested  for  divergence.     If  a 
sev'es  of  positive  terms  already  knoiun  to  be  divergent 

"0  +  «i  + (/8) 

can  be  found  vjhose  terras  are  never  greater  than  the  corresp)onding 
terms  in  the  series  to  be  tested  (a),  then  (a)  is  a  divergent  series. 
Examjjles. 


§§  8,  9.  CONVEUGENCE.  9 

14-l-f-  +  -+ p<l 

This  last  series  can  be  proved  divergent  by  reference  to  the  series 

2  ^  4  ^  G  ^ 
Let  ^—  2  +  4  ■+■  6  + +  2^ 

=K^+^^ +^ 

The  series  in  parenthesis  is  tlie  harmonic  series,  and  its  sum  in- 
creases without  limit  as  n  increases ;  hence  .s„  increases  without  limit 
and  the  series  is  divergent. 


'O^ 


9.    Second  Test  for  Convergence.      The  Test-Ratio.    Let  the  series 
to  he  tested  be 

Wo  +  ^'i  + +  ^K  + 

and  form  the  test-ratio 


n 


When  n  increases  without  limit,  this  ratio  will  in  general  approach  a 
definite  fixed  limit  (or  increase  without  limit).     Call  the  limit  t. 
Then  if  t  <^  1  the  series  is  convergent,  if  t  >  1,  it  is  divergent,  if 
T  :=  1  there  is  no  test  : 

Lim    ^K+i  _  ^  ^  1      Convergent; 

n 

"  T  >  1 ,  Divergent ; 


(( 


T 


=  1,   No  Test. 


First,  let  T  <;  1.     Then  as  n  increases,  the  points  corresponding 
to  the  values  of  ii„  +  ^/u„  will  cluster  about  the  point  t,  and  hence  if 

0  r       y     I 

— I \ 1 Y— 

Vu;.  4. 

a  fixed  point  y  be  chosen  at  pleasure  between  t  and  1,  the  points 
w«  +  i/'^i  ""'ill^  fo^'  sutiiciently  large  values  of  n,  i.e.  for  all  values  of  n 
equal  to  or  greater  than  a  certain  fixed  number  m,  lie  to  the  left 
of  y,  and  we  shall  have 


10  CONVERGENCE.  §  9. 


x<. 

w  ^  m ; 

or, 

n  =  m, 

X  <  ^• 

^™+i  <  ^^y» 

n  =1  m  -{-  1, 

%i + 1 

M„,+2  <  w„,+iy  <  w„,y', 

n  =z  711  -\-  2, 

w„+3<  w«  +  2y  <  «„.y', 

<  ^^„  (y  +  y'  +  y'  + +  yO  <  ^„  p 


Adding  p  of  these  inequalities,  we  get 

J^ 

—  y 

Tlie   sum   of   the   terms   ?«,   beginning  with   ?<,„  +  i   never,   therefore, 

y 

rises  as  high  as  the  value  u^  — - —  •     Hence  the  w-series  converges. 

1  —  y 

The  case  that  t  ^  1  (  or  t  :=  oc  )  is  treated  in  a  similar  manner, 
and  may  be  left  as  an  exercise  for  the  student. 

If  T  =  1  there  is  no  test.  For  consider  series  (3).  The  test- 
ratio  is 


«„  (»^  +  1)''  V  ''. 
and  hence  t  ^  1 ,  no  matter  what  value p  may  have.  But  whenp  ^  1, 
(3)  converges;  and  when  i>  <^  1,  (3)  diverges.  Thus  it  appears 
that  T  can  equal  1  lioth  for  convergent  and  for  divergent  series. 

Memark.  The  student  will  observe  that  the  theorem  does  not  say 
that  the  series  will  converge  if  u„^i/u,^  becomes  and  remains  less 
than  1  when  n  increases,  but  that  it  demands  that  the  limit  of 
u„  +  i/u,^  shall  be  less  than  1.  Thus  in  the  case  of  the  harmonic 
series  this  ratio  n/(n  -\-  1)  is  less  than  1  for  all  values  of  n,  and  yet 
the  series  diverges.     But  the  bmit  is  not  less  than  1. 

Examples.     Are  the  following  series  convergent  or  divergent? 


11      21      2      3 

3~'3'5''3'5'7~^ 

1  +  1  +  1  +  1+.    . 


9  92  93 

1-  +  —  4-1 

9100      I       glOO      1       ^lOi 


2 !  3 !  4! 

_  -(-  -:i^  4-  -11.  + 

100    '    100^    '    100^    '  • 


§§10,11.  CONVERGENCE.  11 

10.  A  Further  Test-Ratio  Test.  The  following  test  for  conver- 
gence and  divergence  is  sometimes  useful ;  the  proof  of  the  rule  is 
omitted.     If 

approaches  a  limit,  let  this  limit  be  denoted  by  a-.      Then  the  series 

Uo  +  Wi  -f 

converges,  if  o"  ^  1  (o^  if  a-  =  oo)  ; 

diverges,  if  a  <^  1  (or  if  a-  =  —  oo)  ; 

if  o-  =  1 ,  there  is  no  test. 

Example. 

1-2  ^  3-4  ^ 


nf  l_!l^^A-  >.^  (2n  — l)2n        ^_  4 -f  i 


^n-fA_      /^  (2n  — l)2n        \  

''n  J  V  (2//  +l)(2n  +  2)y'        2  +  ^+(;-)^ 

and  o-  =  2  ;  the  series  converges. 
Test  the  following  series  : 

11-3        1-3-5 
2    '~  2^    '    2'4-6    ' 

^-^m^im^ 

22  —  a  ~  32  —  a    '    4''^  —  a  ^ 

Apply  any  of  the  foregoing  tests  to  determine  the  convergence  or 
the  divergence  of  the  series  on  pp.  45,  46  of  Byerly's  Problems  in 
Differential  Calculus. 

b)    SERIES   WITH   BOTH   POSITIVE   AND   NEGATIVE   TERMS. 

11.  Alternating  Series.  Theorem.  Let  the  terms  of  the  given 
series  \)  he  alternately  x>ositive  and  negative: 

«o  —  "1  +  «j  —  ''3  + ;  (5) 

2)  let  each  u  he  less  than  (or  equal  to)  its  predecessor : 

w„  +  i<«„; 

8)  let  Lim  »,,  =  0. 

n  ^  cc 

Then  the  series  is  convergent. 


12  CONVERGENCE.  §  11. 


The  following  series  may  serve  as  an  example. 

2^3        4 


1.1        1    , 
1-9  +  5^-7+ (6) 


Proof.     Let 

\  =  ^h  —  '^'l-\-^h +  (—  ly-^u^-i 

and  plot  the  points  Si,  Sg,  Sg, Then  we  shall  show  that  the 

points  Si,  S3,  S5, s-2m  +  \i always  move  to  the  left, 

s^     s^   Se  Ua            u;  S5  S3    s, 
—\ \ — \M \^ — \ \ — 

Fig.  5. 

but  never  advance  so  far  to  the  left  as  So,  for  example.     Hence  by 
the  principle  of  §  4  they  approach  a  limit,  Ui : 

TO  ^  00 

Similarly,  the  points  s,^  ^ii  H^ ^-zmi always  move 

to  the  right,  but  never  advance  so  far  to  the  right  as  s^,  for  example ; 
hence  by  the  same  principle  they  also  approach  a  limit,  U2 : 

Lim  .S2,,  z=  U^ . 

TO  =  00 

Finally,  since 

lim  S2„,  +  i  =  lim  .s\,,„  +  lim  ru^  ; 
TO  ^  00  m  ^  <xi  m  =  00 

but  lim  ?<2m  =  0  i  —  hei"e  the  third  hypothesis  of  the  theorem  comes 
into  play  for  the  first  time ;  —  hence 

Ui  =  Uo  , 
or  simply  U.     Thus  s„  approaches  a  limit,  U,  continually  springing 
over  its  limit. 

Ss  S4      Se     U    So        S.3  Si 


Fig.  6. 

Such  is  the  reasoning  of  the  proof.     It  remains  to  supply  the 
analytical  establishment  of  the  facts  on  which  this  reasoning  depends. 

rU'St,  •''2m +  1    _;:;    ^2m  — 1  ^^^t'  ^'Zm   ^   ^2m-2' 

ForSo,„  +  i  =  ?(o  — O'l  — %)— (M2™-3— «2,«-2)  — («2,«-l  — Wo™) 

-^  •^2jn  —  1           v'-2m  —  1  '"2w)  i 

S2,n=  (''O  — Ml)  + +  ("2,«-4  — «2,«-3)  +  (^2^-2  — «2,«-l) 


s. 


+  (^<2,»,-2 M2»-l)  ; 


2m  — 2     1^  V,'-''2rti,  —  2 


and  the  parentheses  are  all  positive  (or  null). 


§§    11,    12.  CONVKIUJEXCE.  13 

^Xt  S2„,  +  l>-«2  1111(1  -v.    <   Si. 

For  S2„.  +  i  =  .Sj,,,  -\-  «2,„  >  «2  +  M2».  >  «2» 

«2m   ==    ■''2m  M    "2,,.   ^   ^1   "2,,,    <C   ^l' 

The  proof  is  now  complete. 


-  +  --- 

32  -r  52      7! 


Q2      I       K2  72      I 


'     +     ' 


log  2        log  3        log  4 

13.  r/^e  Limit  of  Error  in  the  Alternating  Series.  Suppose  it  be 
required  to  find  the  value  of  series  (5)  eoi-reet  to  fc,  say,  to  3  places 
of  decimals. 

For  tills  purpose  it  is  not  enough  to  know  merely  that  tlic  series 
converges,  and  hence  that  enough  terms  can  be  taken  so  that  their 
sum  .s,,  -will  differ  fnjin  the  limit  U  by  less  than  .001,  for  n  might  be 
so  great,  say  greater  than  10,000,  that  it  would  be  out  of  the  question 
to  compute  s„.  And  in  any  case  one  must  know  when  it  is  safe  to 
stop  adding  terms. 

The  rule  here  is  extremely  simple.  The  sum  of  the  first  n  terms 
of  series  (5),  s,^,  differs  from  the  value  of  the  series,  U,  by  less  than 
the  numerical  value  of  the  (n  -j-  l)st  term.  In  other  words,  we  may 
stop  adding  terms  as  soon  as  we  come  to  a  term  which  is  numerically 
smaller  than  the  proposed  limit  of  error. 

For,  consider  Fig.  (fi).  Tlic  transition  from  .s„  to  s^^i  consists  in 
the  addition  to  s^  of  a  quantity  nuinericall}^  greater  than  the  distance 
from  .s^^  to  U.  This  quantity  is  precisely  the  (n  -\-  l)st  tenn  of  the 
series.     Hence  the  rule. 

For  example,  let  it  be  required  to  compute  the  value  of  the  series 

(7) 


1      1     1   1 

3        2   *  3-"^ 

1 

3 

1 
33 

1 
4 

1      1      .    .    . 
•    34  + 

correct  to  three  places  of  decimals 

1. 

(1)    =  .3333 
1  (i)«-.0123 

^  {-iy=  .0008 

1 
■J 

1 

¥ 
1 

TT 

{^y  —  .0556 

(^y  =  .0031 

(|)«  ::^   .0002 

4  {yy  =  .0000 

.05S9 

.3464 

.3464  —  .058lt  =  .2875 


or,  to  3  places,  tlie  vahu'  of  series  (7)  is  .2S8. 


« 


♦  Tlio  4th  place  is  retained  throughout  the  work  to  insure  accuracy  in  the  third 
place  in  tlie  final  result. 


14  CONVERGENCE.  §§   12,   13. 

Examples.     1.   Show  that  the  value  of  the  series  •• 

1_1J^1J^_1    1,1    1 
2        22^~^32^        42*~'5P 

to  three  places  of  decimals  is  .405. 
2.  How  many  terms  of  the  series 

1-1+1-^+. .... 
2^3        4  ^ 

would  have  to  be  taken  that  the  sum  might  represent  the  value  of 
the  series  correct  to  3  places  of  decimals? 

13.    A  General  Theorem.     J.et 

^'o  +  "i  + 

he  any  convergeyit  series  of  positive  and  negative  terms.      Then 

Lim  ^(„  :=  0  . 
More  generally, 

irhere  p  is  any  integer.,  either  constant  or  varying  with  n. 

The  proof  of  this  theorem  flows  directly  out  of  the  conception  of  a 
limit.     Let 

•\  =  ^'o  H-  ^'i  + +  "„-i 

and  plot  the  points  s^,  .S2,  83, Then  what   we  mern  when 

we  say  "s^^  ajyjroaches  a  limit  U"  is  that  there  is  a  point  /  al)Out 
which  the  s^'s  cluster,  as  n  increases.  This  does  not  necessaiily  re- 
quire that  (as  in  the  series  hitherto  considered)  8^  should  ahvny-^  come 
steadily  nearer  to  U,  as  n  increases.  Thus  Sr,  may  lie  further  away 
from  U  than  s^  does.     But  it  does  mean  that  ultimately  t'ne  s^'s  will 

\ 1 1    Mill 1 1 1 — 

Fig.  7. 

cease  to  deviate  from  U  by  more  than  any  arbitrarily  assigned  quan- 
tity, 8,  however  small.  In  other  words,  let  8  be  taken  at  pleasure 
(=z  1/1,000,000,  say)  and  lay  off  an  interval  extending  to  a  dis- 
tance 8  from  U  in  each  direction,  (Jj  —  8,  U -\-  8);  then  for  the 
larger  values  of  n,  more  precisely,  for  all  values  of  n  greater  than 
a  certain  fixed  number  m,  s  will  lie  within  this  interval.  This 
can  be  stated  algebraically  in  the  following  form  : 

U  —  ^  <C  ^n  ^  ^"  ~f"  ^»  when  n.  ^  771. 


§§   18,   14.  CONVEUGENCE.  15 

Having  thus  stated  what  is  meant  by  s^'s  approaching  a  limit  CT", 
we  now  turn  to  the  proof  of  llic  theorem.     The  sum 

If  H  ^  V},  both  .s,^  and  n_^^,  will  lie  in  the  interval  (U —  8,  C^ -\-  8). 
The  distance  between  these  points  is  therefore  less  than  2  8.     Hence 

no  matter  what  vahie  p  may  have.  But  if  u  (piantity  depends  on  n 
and  can  be  made  to  remain  numerically  as  small  as  is  desired  by 
increasing  n,  then  it  approaches  0  as  its  limit,  when  n  =  y:  .  Thus 
the  proposition  is  established. 

It  is  to  be  noticed  that  while  the  condition  Lim  m^  rr  0  is  necessary^ 
if  the  series  is  to  converge,  it  is  in  no  wise  sufficient  for  the  conver- 
gence. Thus  in  the  harmonic  series  (4)  the  general  term  approaches 
0  as  its  limit,  but  still  the  series  diverges.  The  harmonic  series 
however  does  not  satisfy  the  more  general  condition  of  the  theorem  ; 
for  if  we  put  j)  :=.  n, 

''"+""-^+ +  "«+^^-^  =  ,-rjri+;7^>+ +^7+^>2 

and  does  not  converge  toward  0  as  its  limit.     This  fact  affords  a 
now  proof  of  the  divergence  of  the  harmonic  series. 
It  may  be  remarked  that  the  more  general  condition 

Lim  [;/,,  +  »„  +  i  + +  "„  +  ,.-i]  =  0> 

where  j>  may  vary  with  n  in  nuy  icise  ice  choose,  is  a  sutlicient  con- 
dition for  the  convergence  of  the  series.     See  Appendix. 

14.    Convergence.      The  General  Case.     Let 

be  an}^  series  and  let 

-^'o  +  '"i  + 

denote  the  series  of  positive  terms, 

—  ?(\,  —  >r,  — 

the  series  of  negative  terms,  taken  respectively  in  the  order  in  which 
they  occur  in  (a).     For  example,  if  the  »-series  is 

i_i+i_i+ 

2    '    2-        2*    ' 


l(i  CONVERGENCE.  §  14. 

then  the  ti-series  is 

1  -I- 

22    I    2^ 


1    +   ^.  +    ^4  + 


and  the  —  to-series  is 

_1_1_1_ 

2        2^        2^ 

Let  s^=  Uo-\-  Vi-\- -\-  w„_i, 

«■,«=  ^'o  +  ''i  + +  %n-i^ 

^  =  ic^ -\-  IVi  -\- +  ?«^_i, 


T 


Then,  whatever  value  n  may  have,  5^  can  be  written  in  the  form 

•5«   =    <^,n   —  '^p- 

Here  m  denotes  the  number  of  positive  terms  in  s  ,  o-  their  sum, 
etc.  When  n  increases  without  limit,  both  m  and  p  increase  without 
limit,  and  two  cases  can  arise. 

Case  I.     Both  a^^^  and  t^,  approach  limits  : 

Lim  o-„^  =  V,  Lim  t^,  =:  W; 

m   ^   OO  jp   :^   00 

so  that  both  the  r-series  and  the  t«-series  are  convergent.  Hence 
the  u-series  will  also  converge, 

Lim  8^  =  U, 
n  ^  <xi 
and 

U=  V—  W. 

The  above  example  comes  under  this  case.  Case  I  will  be  of 
principal  interest  to  us. 

Case  II.  At  least  one  of  the  variables  o-,,^,  t^,  approaches  no  limit. 
For  example,  suppose  the  w-series  were 

1,1        1,1        1,1 

'  -  5  +  3  -  i  +  3  -  6  +  7  - 

As  these  examples  show,  the  w-series  may  then  be  convergent  and 
it  may  be  divergent. 

Exercise.  Show  that  if  the  n-series  converges  and  one  of  the 
V-,  w-series  diverges,  the  other  must  also  diverge. 

Let  us  now  form  the  series  of  absolute  values*  of  the  terms  of  the 

*  By  the  absolute  value  of  a  real  number  is  meant  the  numerical  value  of  that 
number.  Thus  the  absolute  value  of  —  H  is  3 ;  of  2^  is  24-  Graphically  it 
means  the  distance  of  the  point  representing  that  number  from  the  point  0. 


§§  14,  15.  CONVERGENCE.  17 

w-series  and  write  this  series  as 

"o  4-  '<'i  H- 

u'^  will  be  a  certain  v,  if  «,,  is  positive ;  a  certain  w,  if  u^  is  negative. 
If  we  set 

it  is  clear  that 


•s'..  =  o- 


+  Tp. 


From  this  relation  wc  deduce  at  once  that  in  Case  I  the  ?t'-serie8 
is  a  convergent  series. 

Conversely,  if  the  u'-series  converges^  then  both  the  v-series  and 
the  ic-series  converge,  and  tee  have  Case  I. 

For  both  the  t'-series  and  the  ?r-serie8  are  series  of  positive  terms, 
and  no  matter  how  many  terms  be  added  in  either  series,  the  sum 
cannot  exceed  the  limit  U'  toward  which  s'  converges.  Hence  bv 
the  piinciple  of  §  4  each  of  these  series  converges. 

Definition.  Series  whose  absolute  value  series  are  convergent 
(i.  e.  M-series  wdiose  '/'-series  converge)  are  said  to  be  absolutely  or 
unconditionally  convergent ;  other  convergent  series  are  said  to  be 
not  absolutely  convergent  or  conditionally  convergent.  The  reason 
for  the  terminology  unconditionally  and  conditionally  convergent  will 
appear  in  §  34. 

15.  Test  for  Convergence.  Since  the  u-serios  surely  converges 
if  the  ?t'-series  converges  —  it  is  then  absolutely  convergent  —  and 
since  the  w'-series  is  a  series  made  up  exclusively  of  positive  terms, 
the  tests  for  convergence  obtained  in  I.  a)  can  be  applied  to  the 
w'-series  and  from  its  convergence  the  convergence  of  the  ^-series  can 
thus  be  inferred.  The  series  that  occur  most  frequently  in  elementary 
analysis  either  come  under  this  head  and  can  be  proved  convergent 
in  the  manner  just  indicated,  or  they  belong  to  the  class  of  alternat- 
ing series  considered  in  §11. 

The  test  of  §  9  can  he  thrown  into  simpler  form  whenever  the  test- 
ratio  "„  +  i/»„  approaches  a  limit,  t\  the  rnle  being  that  when  t  is 
numerically  less  than  7,  the  series  converges  absolutely;  u:hen  t  is 
numerically  greater  than  1,  the  series  diverges ;  when  t  is  numerically 
equal  to  1,  there  is  no  test: 


lim  'i!i±i  —  t 

7i  =  00      ^<„ 


—  1  <^  ^  <^  1  ,  Convergence; 

t  ^  I  or  t  <^  —  1  ,  Divergence; 

t  =  \   ov  t  =  —  \  ,  Xo  Test. 


18  CONVERGENCE.  §§   15,   16. 

For,  the  test-ratio  i<„  +  i/?<„  is  numerically  equal  to  the  test-ratio  of 
the  series  of  absolute  values,  u\^  +  i/u'^.  Now  if  a  variable  /(w) 
approaches  a  limit,  H^  when  n  ^  cc ,  its  numerical  value,  being  the 
distance  of  the  point  representing /(?i)  from  the  point  0,  approaches 
a  limit  too,  namely  the  numerical  value  of  if  (distance  of  if  from  0). 
Hence 

lira    ^'«  +  i 

1 =    "^5 

M  =  CO      u\^ 

where  t  equals  the  numerical  value  of  t.  If  then  —  1  <:^  ^  <^  1,  it 
follows  that  T  <^\  and  the  ^t'-series  converges.  The  w-series  is 
then  absolutely  convergent. 

The  second  part  of  the  rule  will  be  proven  in  the  next  paragraph. 

Example.     Consider  the  series 


x"- 
2 

x^ 
3 

X* 

"I 

+   • 

X 

'^n  +  l 

— 

r^n  +  l 

n 

zzz  _ 

1 

Un 

n  -f-  1 

1 

__  1 

n 

Lim 

^*„+l  _ 

-  t 



-  X 

• 

X 


Hence  the  series  will  converge  when  a-  is  numerically  less  than  1, 

i.  e.  when 

-  1  <  a-  <  1 . 

When  X  =1  1  or  —  1 ,  this  test  fails  to  give  any  information  concern- 
ing the  convergence  of  the  series.  But  it  is  then  seen  directl}'  that  in 
the  first  case  the  series  is  convergent,  in  the  second  case,  divergent. 

16.    Divergence.     To  establish  the  divergence  of  a  series 

^0  +  "i  + 

with  positive  and  negative  terms,  it  is  not  enough  to  establish  the 
divergence  of  the  ^'-series,  as  the  example  of  the  series 

1-1+1-1+ 

shows.  It  will  however  suffice  to  show  that  the  terms  do  not  approach 
0  as  their  limit,  and  this  can  frequently  be  done  most  conveniently 
by  showing  that  the  terms  of  the  w'-serles  do  not  approach  0. 

Thus   if         «  >  1  or  ^  <  —  1 ,        then         t  >  1 

u' 
and  "l*"^  ">  1,  when  w  >  m. 


§§  16,  17.  CONVERGENCE.  19 

Hence  w'„.^i  >  m',„, 

^^'-.  +  2>    "',,+  1    >^^'„., 
<,.  +  3    >«'„,  + 2    >m'„, 

or  u'„  >  «'„,,  n  >  m ; 

that  is,  all  the  u'^'s  from  n  :=  m  on  are  greater  than  a  certain  positive 
quantity  p  =  w',,^  and  hence  u\^  and  ?<^  cannot  approach  0  as  their 
limit,  when  n  ^  co. 

Example.  In  the  series  of  §  15,  <  =r  —  x-;  hence  this  series  di- 
verges for  all  values  of  x  numerically  greater  than  1.  These  results 
may  be  represented  graphically  as  follows  :  — 

Divergent — 1  0 1  Divergent 

Conrergmt 

Exercise.  For  what  values  of  x  are  the  following  series  conver- 
gent, for  what  values  divergent  ?  Indicate  these  values  by  a  diagram 
similar  to  the  one  above. 

x"^  x^ 

Ans.  —  1  ^  a;  <^  1 ,  Conv.  ;  x^\^x<^  —  1 ,  Div. 

x^    ,    x^        x''    , 

^-3  +  5-7  + 

l+x-  +  ^  +  3^+ 

10a;  +  102.1-2  +  10«.»-3  -|- 

1  +  .r  -|-  2  !  a-2  -|-  3  !  x^  -\- 

17.    Theorem.     Let 

«0   +   «1    +    «2   H- 

be  any  ahsoluteUj  convergent  series;  po^  pi-  p>i (^^^y  set  of 

quantities  not  increasing  numerically  indejinitely.      Then  the  series 

converges  absolutely. 

For,  let  a',,,  p\^  be  tlie  aiisolute  vahics  of  k^^,  p_^  respectively,  fl  a 
positive  quantity  greatt'r  than  any  of  the  q^iantities  p\^,  and  form  the 
series 

«'op'o  +  <'\p\  +  ^f'.p'.  -h 


20  CONVERGENCE.  §§  17,  18. 

The  terms  of  this  series  are  less  respectively  than  the  terms  of  the 
convergent  series 

and  each  series  is  made  up  exclusively  of  positive  terms.     Hence  the 
first  series  converges  and  the  series 

converges  absolutely. 

Examples.     1.  The  series 

sin  a;        sinS.i'        sin  5  a; 

12  g2        "I  T2 

converges  absolutely  for  all  values  of  x.     For  the  series 

converges  absolutely  and  sin  nx  never  exceeds  unity  nmnerically. 

2.  If    Oq  +  «!  +  «o  + and    &i  +  &2  + are 

any  two  absolutely  convergent  series,  the  series 

f'o  ~\-  f^i  cos  X  -\-  tto  cos  2.1;  4- 

and  bi  sin  x  -(-  63  sin  2  a;  -|- 

converge  absolutely. 

3.  Show  that  the  series 

e~^  cos  X  -|-  e^^^  cos  2x  -\- 

converges  absolutely  for  all  positive  values  of  x. 

4.  What  can  j'ou  say  about  the  convergence  of  the  series 

1  +  r  cos  ^  -|-  r^  cos  2  ^  + ? 

18.    Convergence  and  Divergence  of  Power  Series.     A  series  of 
ascending  integral  powers  of  a,', 

a,,  -\-  a^x  -\-  a.x-  -f- , 

where  the  coefficients  Oq,  «i,  «25 ^^'^  independent  of  x,  is 

called  a  jwiver  series.  Such  a  series  may  converge  for  all  values  of 
X,  but  it  will  in  general  converge  for  some  values  and  diverge  for 
others.  In  the  latter  case  the  interval  of  convergence  extends  equal 
distances  in  each  direction  from  the  point  x  =  0,  and  the  series  con- 
Divergent  —  r 0 r  Divergent 

Convergent 

verges  absolutely  for  every  point  x  lying  ivithin  this  interval,  but  not 
necessarily  for  the  extremities  of  the  interval. 


§   is.  CONVERGENCE.  21 

Tlie  proof  is  as  follows.  Let  Xq  be  any  value  of  x  for  which  the 
terms  of  the  power  series  a^x^  do  not  increase  without  limit ;  a'^,  x'^, 
the  absolute  values  respectively  of  a„,  x„.  Then  a'^x',"  is  less  than 
some  fixed  positive  quantity  C,  independent  of  j(,  for  all  values  of  n. 
For  X  =^  a'o,  the  power  series  may  converge  and  it  may  diverge. — 
Let  h  be  any  value  of  x  numerically  less  than  x'q  ;  h'  its  numerical 
vahic.     Then  the  power  series  converges  absolutely  for  x  =  h.     For 

a\^''^  =  a'^xV  C'yX  <  Cr" , 

wdiere  r  =  Ji'/x'q  <^  1.     Hence  the  t(Mnis  of  the  absolute  value  series 

a'oH-  a\h'  -\-  <i',h''-\- 

are  less  respectively  than  the  terms  of  the  convergent  geometric  series 

C  +  Cr  -\-  Cr^  -f- 

and  the  series  • 

converges  absolutely. 

From  this  theorem  it  follows  that  if  the  power  series  converges  for 
X  =.  .To,  it  converges  absolutely  for  all  values  of  x  within  an  interval 
stretching  from  0  to  x^  and  reaching  out  to  the  same  distance  on  the 
other  side  of  the  point  a;  =  0  ;  and  if  diverges  for  x  =  x^,  it  diverges 
for  all  values  of  x  lying  outside  of  the  interval  from  Xi  to  —  x^.  If 
now  the  series  ever  diverges,  consider  the  positive  values  of  x  for 
which  it  diverges.  They  fill  a  region  extending  down  to  a  point 
X  z=  r,  where  r  in  general  is  greater  than  0  and  such  that  the  series 
converges  absolutely  for  all  values  of  x  numerically  less  than  r ;  and 
this  is  what  was  to  be  proved. 

A  simpler  proof  of  this  theorem  can  be  given  for  the  special  case 
that  a„  +  i/a„  approaches  a  limit,  L,  when  n  =:  x)  .     For  then 

Lini     "«  +  i  lim     ^^n  + 1  •^"      __   r^ 

TC  =  00     u^  71  =  CO       a,,  x"  ' 

or  t  =:  Lx.  Hence  when  Jj  =  0,  the  power  scries  converges  abso- 
lutely for  all  values  of  x*  (H'J)  ;  while  if  L  dr  0,  the  series  converges 
absolutely  when  x  is  numerically  less  than  \  L,  and  diverges  when  x 
is  numerically  greater  than  1/L.     This  proves  the  proposition. 


II.  SERIES  AS  A  MEA^S  OF  COMPUTATION. 


«)    THE   LOGARITHMIC   SERIES. 

19.  One  of  the  most  important  applications  of  infinite  series  in 
analysis,  and  the  one  that  chiefly  concerns  us  in  this  course,  is  that 
of  computing  the  numerical  value  of  a  complicated  analytic  expres- 
sion, for  example,  of  a  definite  integral  like 


i: 


,— x2 


dx  , 


when  the  indefinite  integral  cannot  be  found.  In  fact,  the  values  of 
the  elementary  transcendental  functions,  the  logarithm,  sine,  cosine, 
etc.,  are  computed  most  simply  in  this  way.  Let  us  see  how  a  table 
of  logarithms  can  be  computed  from  an  infinite  series. 

A  series  for  the  function  log^  (1  -|-  h)  can  be  obtained  as  follows. 
Begin  with  the  formula 


log(l  +  /0 


_   r'^  cix 

~  Jo  rr 


X 

The  function  (1  -{-  x)~'^  can  be  represented  by  the  geometric  series 

=:  1  X  -\-  .^■^  —  x^  -\- 


1  +  x 
Integrate  each  side  of  this  equation  between  the  limits  0  and  h  : 

— ^- —  ^    I     \  •  dx  —    I     xdx  -\-    I    x'^dx  — 

0    1  i"  -^^        Jo  Jo  Jo 

Evaluating  these  integrals  we  are  led  to  the  desired  formula : 

log(l-h70==/^-2-+  3- (8) 

In  deducing  the  above  formula  it  has  been  assumed  that  the  theo- 
rem that  the  integral  of  a  sum  of  terms  is  equal  to  the  sum  of  the 
integrals  of  the  terms  can  be  extended  to  an  infinite  series.  Now 
an  infinite  series  is  not  a  sum,  but  the  limit  of  a  sum,  and  hence  the 
extension  of  this  theorem  requires  justification,     v.  §§39,  40. 


§§   11),  20.        SERIES    AS    A    MEANS    OE    COMPUTATION.  23 

Exercise.     Obtain  the  formula 

tan    ^  h  :=  h r  +   ^ 

;3  5 

Hence  evaluate  the  series 

'-a  +  o- 

20.    In  the  examples  of  §  12  the  value  of  series  (8)  was  computed 

to  three  places  of  decimals  for  h  =:  ^  and  //  =r  i  ;   and  it  thus  appears 

that 

log  li  =  .287  (5),  log  U  =  .405  (5). 

To  find  log  2  we  could  sulistitute  in  («)  the  value  h  ^  1  : 

But  this  series  is  not  well  adai)ted  to  numerical  computation.*  In 
fact  to  get  the  value  of  log  2  correct  to  the  third  place  of  decimals,  it 
would  be  necessary  to  take  1000  terms.  A  simple  device  however 
makes  the  computation  easy.     Write 

•>  —   4  .  3 

-    3       ? 

and  then  take  the  logarithm  of  each  side : 

log  2  =  log  A  +  log  I 

=  .287  (;-))  -f-  .405  (5)  =  .693  (O); 

Hence,  to  three  places,  log  2  =^  .693. 

Next,  to  find  log  5.  Here  the  series  must  be  applied  in  still  a 
different  way,  for  if  1  -{-  h  be  set  equal  to  5,  /i  ==  4,  and  the  series 
does  not  converge.     We  therefore  set 

5  =  4+  1  =4(1  +i), 

log  5  =r  2  log  2  -|-  log  IJ 

=:  1.386  (0)  +  .223  (2)  =  1.609  (2), 

where  log  1^  is  computed  directly  from  formula  (8). 

From  the  values  of  log  2  and  log  5,  log  10  can  at  once  be  found. 

log  10  =  log  2  4-  log  5  =  .693  (0)  -f  1.609  (2)  =  2.302  (2) 

or  to  3  places. 

This  latter  logarithm  is  of  great  importance,  for  its  value  must  be 
known  in  order  to  compute  the  denary  logaritlim  from  the  natural 

*  The  formula  is  nevertheless  useful  as  showing  the  value  of  a  familiar 
series,  (0).  We  could  not  find  by  direot  conipntation  the  value  of  this  series  to, 
say,  seven  places,  because  the  work  would  be  too  long. 


log  10  =  2.302. 


24  SERIES    AS    A    MEANS    OF    COMPUTATION.         §§  20,  21. 

logarithm.     By  the  formula  for  the  transformation  of  logarithms 
from  the  base  c  to  the  base  ?>, 

,         .        log,  J. 

we  have  ^^S- ^  =  log:iO  ' 

Hence  for  example 

.693 

">«-'  =  2m  =  ■'''■ 

Examples.     Compute 

log  20,  logio20, 

log  9,  logio9, 

log  13,  logiol3.  .      ■ 

21.  Series  (8)  is  thus  seen  to  serve  its  i>uipose  well  when  only  a 
few  places  of  decimals  are  needed.  Suppose  however  we  wished  to 
know  log  2  correct  to  7  places  of  decimals.  Series  (8)  would  then 
give  less  satisfactory  results.  In  fact,  it  would  require  16  terms  of 
the  series  to  yield  log  li  to  7  places. 

From  (8)  a  new  series  can  be  deduced  as  follows.  Let  h  z=  —  x. 
Then  (8)  becomes 

log  (l—x)  =  —x—-  —  ~— 

Next  replace  h  in  (8)  by  x : 


x'-    ,    x^ 


log  (1  +.«)  =  +  X  —  -  +  -  — 

Subtracting  the  former  of  these  series  from  the  latter  and  combining 
the  logarithms  we  get  the  desired  formula : 


iJi5=  ,  (,  + 1  +  ^' 


We  have  subtracted  on  the  right  hand  side  as  if  we  had  sums. 
We  have  not ;  we  have  limits  of  sums.  This  step  will  be  justified 
in  §35. 

"We  will  now  apply  series  (9)  to  the  determination  of  log  2  to  seven 
places.     X  must  be  so  chosen  that 

~ — — ;  =.  2,  1.  e.  ^  =^  ^  and 

,     1  +  i        /I  ,  1  1   ,  1  1   ,  1  1   ,  \ 

^^^1-^=1=^3  +  3   3^  +  5   3^+7   3^+ ) 


§   21.  SEIilKS    AS    A    MHANS    l)V    C(XMl'LTATION.  25 

The  advantage  of  this  series  over  (8)  is  twofold  :  first,  it  suflices  to 
compute  tlie  value  of  the  series  for  one  value  of  .r,  x  =  ^,  and 
second,  the  series  converges  more  rapidly  than  (H)  for  a  given 
value  of  X,  since  only  the  odd  powers  of  x  enter. 


(i)  = 

.333 

333 

33 

(i)  = 

.333 

333 

33 

G)«  = 

.037 

037 

04 

1 

7 

■  ar  = 

.012 

345 

68 

(hr  = 

.004 

115 

23 

1 

■  i-hr  = 

.000 

823 

05 

a)'  - 

.000 

457 

25 

1 
T 

■  ar  = 

.000 

065 

32 

ar  = 

.000 

050 

81 

1 

■g- 

■  (hr  = 

.000 

005 

65 

(i)"  = 

.000 

005 

65 

1 
11 

•(^)"  = 

.000 

000 

51 

(^y  = 

.000 

000 

G3 

1 

■(iy'  = 

.000 

000 

05 

(h''  = 

.000 

000 

07 

1 
TZ 

■(hy'  = 

.000 

000 

00 

.346     573     59 

The  term  -j^  (^y^  has  no  effect  on  the  eighth  decimal  place.  But 
this  is  not  enough  to  justify  us  in  stopping  here.  We  must  show 
that  the  remainder  of  the  series  from  this  point  on  cannot  influence 
this  place  either.     Now  the  remainder  is  the  series 

15   3^6  ^  17   3"  ^  19   318  ^ 

15   315  L  17   32^  19   3*^  J 

The  value  of  the  series  in  brackets  cannot  be  readily  determined ; 
nor  is  tliat  necessary,  for  it  is  obviously  less  than  the  value  of  the 
series  obtained  from  it  ])y  discarding  the  coefiicients  ^,  ^|,  etc., 
i.  e.  than  the  geometric  series 

,      1,1,  19 


3'^    '    3*    '  1  —  I        8 

and  hence  the  remainder  in  question  is  less  than 

J_    J^     9 
15     3^5    8' 

and  so  does  not  affect  the  eighth  place. 
We  obtain  then  finally  for  log  2  the  value 

2  X  .346     573     5(9)  =.693     147     1(8) 

or  to  seven  places 

log  2  =  .693  147  2. 


26  SERIES    AS    A    MEANS    OF    COMPUTATION.         §§  21,  22. 

Examples.     Show  that 

log  li  =     .223     143     (4) 

log  5    =  1.609     437      (8). 

Compute  log  2  by  aid  of  the  formula 

log  2  =  —  log  ^  =  —  log  I  —  log  |. 

Knowing  log  2  and  log  5  we  can  find  log  10  : 

log  10  =  2.302     585. 

Exam2)le.     Compute 

logio  2  ,  logio  9 

to  six  places. 

Series  (9)  is  thus  seen  to  be  well  adapted  to  the  computation  of 
logarithms.  If  y  denote  any  positive  number  and  x  be  so  deter- 
mined that 

1  +  .^;  .  .V  —  1 

— ! —  =  y  ,     I.e.     X  =1  — , — -  , 

then  X  always  lies  between  —  1  and  -j-  1  and  series  (9)  converges 
towards  the  value  log  y.  For  values  of  y  numerically  large  the 
convergence  will  be  less  rapid  and  devices  similar  to  those  above 
explained  must  be  used  to  get  the  required  result. 

In  the  actual  computation  of  a  table,  not  all  the  values  tabulated 
are  computed  directly  from  the  series.  A  few  values  are  computed 
in  this  way  and  the  others  are  found  by  ingenious  devices. 

b)    THE   BINOMIAL    SERIES. 

22.    In  elementary  algebra  the  Binomial  Theorem  for  a  positive 

integral  exponent : 

7)z  ( ')yh 1 1 

(a  +  hy  =  a'"  +  ma'"-i6  -] ^—-^ — La'^-'^h^  -\- 

i.    •    ^ 

(to  m  -\-  1  terms) 
is  established. 

Consider  the  series 

1  -h  /^^  +        1.2  ^  1  •  2  •  3  ^ 

If  /a  is  a  positive  integer,  this  series  breaks  off  with  /a  +  1  terms,  for 
then,  from  this  point  on,  each  numerator  contains  0  as  a  factor. 
Thus  if  /A  r=  2,  we  have 

2-1  2  •  1  •  0 

1  _j_  2x  -^ x^  -\ — -  x^  -f"  6tc.  (subsequent  terms  all  0), 


§  'I'l.  SERIES   AS   A    MEANS   OF   COMPUTATION.  27 

or  simply  1  -\-  'Ix  -\-  x^.  In  this  case  the  series  is  seen  by  compari- 
son with  the  binomial  formula  (a  :=  I,  h  z=  x,  m  z=z  fj.)  to  have  the 
value  (I  -\-  x)f: 

\      I      ''  ^  ^     ^       1  ■  2  '  1-2-.3  ' 

If  however  /a  is  any  nuinlter  not  a  positive  inte<!;er  or  zero  (negative 
number,  fraction,  etc.)  the  series  never  breaks  olf,  i.e.  it  becomes  an 
infinite  series.  Let  us  see  for  what  vahies  of  x  it  converges,  for 
only  for  such  values  will  it  have  a  meaning.  The  general  term  of 
the  series  is 


Hence 


fxjfx—  1)  {fi.  —  2) (fi  —  n  -\-  \) 

1-2         .       3- •      n 


H-JH-—^) {/^  —  n  -\-  1)  (i^  —  n)  ^^^^ 


u^  +  1  1     •    2    • n      •      {n  -{-  I) 

w„    ~  fj.(ti—l) (/x  — n+  1)  ^ 

1    •    2 '.       n  "^ 

a  —  n  1  —  a/n 

"~  n  4-  1      ~        l  +  l/n 
and 

Lim    ^^,  + 1  __  ^  ^ 

n  =  oa      u 

n 

Consequently  the  series  converges  for  all  values  of  x  numerically  less 
than  unity.  (§  15.)  For  the  values  x  =  1,  —  1  special  investiga- 
tion is  necessary,  which  we  will  not  go  into  here. 

Divergent  — 1 0 T Direr genl 

Convergent 

We  may  note  in  passing  that  when  0  <^  x*  <^  1  the  series  finally 
becomes  an  alternating  series,  a  fact  that  is  useful  when  the  series 
is  used  for  computation. 

Toward  what  value  does  the  series  converge  when  x  lies  between 
—  1  and  -|-  1  ?  The  answer  to  this  question  is  as  follows  :  For  all 
values  of  x  for  which  (he  binomial  series  converges,  its  value  is 
(1  +  xr : 

(1  +  xy  =  1  4-  ^.r  +  ^^^~^^  x'-\- (10) 

The  proof  of  this  theorem  will  not  be  considered  here  (v.  Chap. 
III).  Let  us  first  see  whether  the  series  is  of  any  value  for  the  pur- 
poses of  computation. 

Example  I.  Let  it  be  required  to  compute  <^  35  correct  to  five 
places. 


28  SERIES   AS    A    MEANS   OF    COMPUTATION.  §  22. 

We  must  throw  the  radicand  into  a  form  adapted  to  computation 
by  the  series.     We  do  this  as  follows.     Since  2^  =  32  we  write 

35  =  32  +  3  =  2^1  +^), 

^35  =  2-  (1  +  3%)^- 
The  second  factor  can  be  computed  by  aid  of  the  series. 

=  1  -f  .018  750  —  .000  703  +  .000  040  —  .000  003 
=  1.018  08(4) 
and  -^  35  =  2.036  17. 

Exercise.     Show  that  in  the  above  computation  we  are  justified  in 
breaking  off,  as  we  did,  with  the  fifth  term. 
Example  II.     Find  s/  15  to  five  places. 
Here  we  have  a  choice  between  the  expressions 

15=     8+     7  =  2^1  +  1) 

and  15=27  —  12  =  3^(1—1) 

In  the  first  case  (1  -\-  |^)j,  in  the  second  (1  —  |)s  would  be  com- 
puted by  aid  of  the  series.  In  practice  however  there  is  no  question 
as  to  which  expression  to  use,  for  the  second  series  converges  more 
rapidly  than  the  first. 

Examples.     1.  Complete  the  computation  of  -^  15. 

2.  Show  that       ^  9  =  2.080  09       and       -</  2000  =  2.961  94. 

3.  Compute        \J  2        first  by  letting  /a  :=  —  |^,  a;  =  —  |^ ; 
then  by  writing  2  =:  f  •  f . 

4.  Find  ^  2  to  five  places  by  any.  method. 

5.  Obtain  from  (10)  the  following  formulas  : 

1 


1 

1 

X 

(1 

1 

xy 

V 

1  — 

-x^ 

=  1  —  -^  -\-  '^^  —  '^^  -l- 


—  1  _  2.r  -|-  3.t2  —  4.c3  -(- 


i  +  ^-'  +  ^-'  +  m-'^'  + 


Vi-.»=i_i.»_^_.._^.. 


§§  23,  24.        SERIES    AS    A    MEANS    OF    rOMi'LTATIOX.  2i> 

23.  Series  for  sxn~'^h  and  tan~^li.  Tlic  Co mputation  of  n. 
The  method  set  forth  in  §  19  is  applicable  to  the  representation  of 
8in~^/i  and  tan~^/i  (v.  Exercise,  §  19)  by  series. 

Jr*h        dx  1   /<3         1.3    7,6 

tan-i/(  =  h—       -\-  y  — (12) 

From  these  series  the  value  of  ir  can  be  computed.  If  in  series 
(12)  wo  set  /;  r=  1,  we  get  the  equation  : 

4  3^5        7  ^ 

This  series,  like  series  (6),  is  not  well  adapted  to  computation.  A 
better  series  is  obtained  by  putting  h  ^  ^  \\\  series  (11)  : 

6        2    '    2    ;}  \'>)     '2-4    f)  \->)    ^ 

This  series  yields  readily  three  or  four  places  of  decimals;  but  if 
greater  accuracy  is  desired,  more  elaborate  methods  are  necessary, 
(v.  Jordan,  Cours  d' Analyse^  Vol.  I,  §  2r)2  ;   1893). 

Exercise.  If  the  radius  of  the  Earth  were  exactly  4000  miles,  to 
how  many  places  of  decimals  should  you  need  to  know  v  in  order  to 
compute  the  circumference  correct  to  one  inch?  Determine  ir  to  this 
number  of  places  by  Jordan's  method. 

24.  The  Length  of  the  Arc  of  an  Ellipse.  Let  the  equation  of 
the  ellipse  be  given  in  the  form  : 

X  ^=  a  sin  <^  ,  y  =  b  cos  <^ . 

Then  the  length  of  the  arc,  measured  from  the  end  of  the  minor  axis, 
will  be 

VI  —  e^  sin-^  <f>   dcji , 
0 


s 


where  (a^  —  b'^)/a-  =  e^  <^  1.  The  integral  that  here  presents  itself 
is  known  as  an  Elliptic  Integral  and  its  value  cannot  be  found  in  the 
usual  way,  since  the  indefinite  integral  cannot  be  expressed  in  terms 
of  the  elementary  functions.  Its  value  can  however  be  obtained  by 
the  aid  of  infinite  series.  The  substitution  of  esin<^  for  x  in  the 
last  example  of  §  22  gives  the  formula 

V  1  —  e^  sin^  0  =  l  —  -e^  sin^  <^  —  — —  e*  siu^  </>  — 


30  SERIES    AS    A    MEANS    OF    COMPUTATION.         §§  24,  25. 

Hence  (v.  §  40) 

s=z  aTcfi  —  ^e^  f'^sm^cj>dcf>  —  ^e*  I    sm'<j>d<i> 1 

These  integrals  can  be  evaluated  by  the  aid  of  the  formulas  of  IV  of 
Peirce's  Short  Table  of  Integrals.  In  particular,  the  length  of  a 
quadrant  S  will  be  found  by  putting  (ji  =  ^  n  and  using  the  formula 
(No.  483  of  the  Tables,  1899,  or  later,  edition) 


TT 

J~- 


1  •  3  •  5 {n~  1)    TT 

sin"  d)  dA  =  - — - — ^ ^  ,  n,  an  even  integer. 

0  2-4-6 n  2 


The  elliptic  integral  then  becomes  the  integral  known  as  the  Complete 
Elliptic  Integral  of  the  Second  Kind ;  it  is  denoted  by  E  : 

TT 

E  =    j^  1  —  e^sin-^c^  r?<^  . 


(No.  248  of  the  Tables). 

-  aE . 

If  e  =:  0  the  ellipse  reduces  to  a  circle  and  S  =  ^-na. 

Examples.  1.  Compute  the  perimeter  of  an  ellipse  whose  major 
axis  is  twice  as  long  as  the  minor  axis,  correct  to  one  tenth  of 
one  percent. 

2.  A  tomato  can  from  which  the  top  and  bottom  have  been  removed 
is  bent  into  the  shape  of  an  elliptic  cylinder,  one  axis  of  which  is 
twice  as  long  as  the  other.  Find  what  size  to  make  the  new  top  and 
bottom.  If  the  original  can  held  a  quart,  how  much  will  the  new 
can  hold  ? 

25.  The  Period  of  Oscillation  of  a  Pendulum.  It  is  shown  in 
Mechanics  (v,  Byerly's  Int.  Cal.,  Chap.  XVI)  that  the  time  of  a 
complete  oscillation  of  a  pendulum  of  length  I  is  given  by  the  formula 


\9  Jo    V  1  —  k''sm'<l> 


sin-, 


where  a  denotes  the  initial  inclination  of  the  pendulum  to  the  vertical. 
K  is  known  as  the  Complete  Elliptic  Integral  of  the  First  Kind  and 
its  value  is  computed  as  follows.  The  substitution  of  Z:sin<^  for  a; 
in  the  series  for  (1  —  a;^)~i  gives  the  formula  (v.  Exs.,  §  22). 


^1  —  k^.s'm'^<t) 


§§  25,  2fi.        SERIES    AS    A    MEANS    OF    ('(JMl'LTATIOX.  31 

Integrating  and  reducing  as  in  §  24,  we  obtain  the  formula 

If  the  angle  through  which  the  pendulum  oscillates  is  small,  an 
approximation  for  T  suflicicntly  accurate  for  most  purposes  will  be 
obtained  by  putting  k  =i  {).     Then  K  ^z  ^tt  and 


--j;. 


the  usual  pendulum  formula. 

Exercise.  Show  that  if  a  <^  5°,  this  approximation  is  correct  to 
less  than  one  tenth  of  one  percent. 

c)    APPROXIMATE    FORMULAS   IN   APPLIED    MATHEMATICS. 

36.  It  is  often  possible  to  replace  a  complicated  formula  in  applied 
mathematics  by  a  simpler  one  which  is  still  correct  within  the  limits 
of  error  of  the  observations.* 

The  Coefficient  of  Expansion.  By  the  coefficient  of  linear  expan- 
sion of  a  solid  is  meant  the  ratio 

where  I  is  the  length  of  a  piece  of  the  substance  at  temperature  ?°,  V 
the  length  at  temperature  t'°.  The  coefficient  of  cubical  expansion 
is  defined  similarly  as 

V  —  V 

/3  =  — p-/(^'-0, 

where  F,  V  stand  for  the  volumes  at  temperature  t°,  t'°  respectively. 
Then 

V  —  V      V^  —  P 


V  I'      ' 

as  is  at  once  clear  if  we  consider  a  cube  of  the  substance,  the  length 
of  an  edge  being  /  at  t°.  The  accurate  expression  for  a  in  terms  of  ^ 
is  as  follows. 


*  See  Kohlrausch,  Phygtca/  .Ueastirenients,  §§  1-G. 


•      •      • 


32  SERIES    AS    A    MEANS    OF    COMPUTATION.         §§  26,  27. 

Since  /3  is  small, — usually  less  than  .0001,  —  the  error  made  by 
neglecting  all  terms  of  the  series  subsequent  to  the  first  is  less  than 
the  errors  of  observation  and  hence  we  may  assume  without  any  loss 
of  accuracy  that 

a  ^  ^  /? ,  )8  ^  3a  . 

Double  Weighhig.  Show  that  if  the  apparent  weight  of  a  body 
when  placed  in  one  scale  pan  is  j)^,  when  placed  in  the  other  scale 
pan,  p.2  (the  difference  being  due  to  a  slight  inequality  in  the  lengths 
of  the  arms  of  the  balance) ,  the  true  weight  p  is  given  with  sufficient 
accuracy  by  the  formula  : 

P  =  i(i?i  +i?2)- 

37.  Errors  of  Observation.  In  an  experimental  determination  of 
a  physical  magnitude  it  is  important  to  know  what  effect  an  error  in 
an  observed  value  will  have  on  the  final  result.  For  example,  let  it 
be  required  to  determine  the  radius  of  a  capillary  tube  by  measuring 
the  length  of  a  column  of  mercury  contained  in  the  tube,  and  weigh- 
ing the  mercury.     From  the  formula 

where  w  denotes  the  weight  of  the  mercury  in  grammes,  I  the  length 
of  the  column  in  centimetres,  p  the  density  of  the  mercury  (=:  13.6), 
and  r  the  radius  of  the  tube,  we  get 


T=     \—,—  .1530 


w 

1530         ly 


Now  the  principal  error  in  determining  r  arises  from  the  error  in 
observing  I.  Let  I  be  the  true  value,  V  ^  I  -\-  e  the  observed  value 
of  the  length  of  the  column ;  r  the  true  value,  r'  ^  r  -\-  E  the  com- 
puted value  of  the  radius.  Then  E  is  the  error  in  the  result  arising 
from  the  error  of  observation  e,  the  error  in  observing  to  being 
assumed  negligible.     Hence 

^  =  --J|---Jf  =  -3jL»((.  +  ;)-'_.) 

_  1    e        3   e\   _   _ 


Since  e  is   small  we  get  a  result  sufficiently  accurate  by  taking 
only  the  first  term  ;  and  hence,  approximately, 

E  =  —^r-~-e' 


§§  27,  28.        SERIES    AS    A    MEANS    OF    COMPUTATION'.  33 

Thus  for  a  given  error  in  obsorvin<>;  /,  the  error  in  the  computed  value 
of  r  is  inversely  proportional  t(;  tlie  length  of  the  column  of  mercury 
used,  —  a  result  not  a  priori  obvious,  for  r  itself  is  inversely  propor- 
tioned only  to  V  '• 

Exercise.  An  engineer  surveys  a  field,  using  a  chain  that  is 
incorrect  by  one  tenth  of  one  percent  of  its  length.  Show  that  the 
error  thus  arising  in  the  determination  of  the  area  of  the  fieM  will  be 
two  tenths  of  one  percent  of  the  area. 

28.  Pendulum  Problems.  A  clock  regulated  Ity  a  pendulum  is 
located  at  a  point  (A)  on  the  earth's  surface.  If  it  is  carried  to  a 
neighboring  point  (B),  h  feet  above  the  level  of  (^-1),  show  that  it 
will  lose  ^5^  h  seconds  a  day,  i.  e.  one  second  for  every  244  feet  of 
elevation. 

The  number  of  seconds  N  that  the  clock  registers  in  24  hours  is 
inversely  proportional  to  the  period  T  of  the  oscillation  of  the  pen- 
dulum.    Hence  (cf.  §25) 

N 


where  the  unpriraed  letters  refer  to  the  location  (A) ,  the  primed  letters 
to  (B).     If  the  clock  was  keeping  true  time  at  (A),  then  N  =i  86,400. 

g-  {R  +  hy' 

where  R  denotes  the  length  of  the  radius  of  the  earth.     (Cf.  Byerly's 
Diff.  Cal.,  §117.)     Hence 

A^—  ^y  :=  .V  (  1  —     ]■-  ]  =  N      '^ 


Mh 


7?  +  7/ 


R  R'  ^       R^ 

If  h  does  not  exceed  4  miles,  h/R  <  .001,  h^/R-  <  .000  001,  and 
the  first  term  of  the  series  gives  N  —  N'  correct  to  seconds  : 

Examples.  1.  The  summit  of  Mt.  AVashington  is  G226  feet  above 
the  sea  level.  How  many  seconds  a  day  will  a  clock  lose  that  keeps 
accurate  time  in  Boston  Harbor,  if  carried  to  the  summit  of  the 
Mountain  ? 

2.  A  pendulum  that  beats  seconds  on  the  surface  of  the  earth  is 
observed  to  gain  one  second  an  hour  when  carried  to  the  bottom  of  a 
mine.  How  deep  is  the  mine?  Assume  the  attraction  at  interior 
points  of  the  earth  to  vary  as  the  distance  from  the  centre. 


34  SERIES    AS    A    MEANS    OF    COMPUTATION.  §  29. 

29.  Exercises.  1.  Show  that  the  correction  for  expansion  and 
contraction  due  to  heat  and  cold  is  given  by  the  formula 

71  =  43,200  a  ^, 

■where  a  denotes  the   coefficient  of  linear  expansion,  t  the  rise  in 
temperature,  and  w  the  number  of  seconds  lost  in  a  day. 

For  brass,  a  =^  .000  019,  t  being  measured  in  degrees  centigrade. 
Thus  for  a  brass  pendulum  »  =  .82  t,  and  a  rise  in  temperature  of 
5°  causes  the  clock  to  lose  a  little  over  4  seconds  a  day. 

2.  A  man  is  standing  on  the  deck  of  a  ship  and  his  eyes  are  h  ft. 
above  the  sea  level.  If  D  denotes  the  shortest  distance  of  a  ship 
away  whose  masts  and  rigging  he  can  see,  Ijut  whose  hull  is  invisible 
to  him,  hi  the  height,  measured  in  feet,  to  which  the  hull  rises  out  of 
the  water,  show  that,  if  refraction  can  be  neglected, 

D  =  1.23  (V  ^^  +  V  ^^i)  miles. 

If  //  z=  /^i  =  16  ft.,  />  =  10  miles  (nearly). 

3.  Show  that  an  arc  of  a  great  circle  of  the  earth,  2 J  miles  long, 
recedes  1  foot  from  its  chord. 

4.  Assuming  that  the  sun's  parallax  is  8". 76,  prove  that  the  dis- 
tance of  the  sun  from  the  earth  is  about  94  million  miles. 

5.  Show  that  in  levelling  the  correction  for  the  curvature  of  the 
earth  is  8  in.  for  one  mile.     How  much  is  it  for  two  miles? 

6.  The  weights  of  an  astronomical  clock  exert,  through  faulty 
construction  of  the  clock,  a  greater  propelling  force  when  the  clock 
has  just  been  w^ound  up  than  when  it  has  nearly  run  down,  and  thus 
increase  the  amplitude  of  the  pendulum  from  2°  to  2°  4'  mi  each  side 
of  the  vertical.  Show  that  if  the  clock  keeps  correct  time  when  it 
has  nearly  run  down,  it  will  lose  at  the  i-ate  of  about  .4  of  a  second 
a  day  when  it  has  just  been  wound  up. 

7.  Two  nearly  equal,  but  unknown  resistances,  A  and  B,  form 
two  arms  of  a  Wheatstone's  Bridge.  A  standard  box  of  coils  and 
a  resistance  x  to  be  measured  form  the  other  two  arms.  A  balance 
is  obtained  when  the  standard  rheostat  has  a  resistance  of  r  ohms. 
When  however  A  and  B  are  interchanged,  a  balance  is  obtained 
when  the  resistance  of  the  rheostat  is  r'  ohms.  Show  that,  ap- 
proximately, 

x=  .H>-+  '•')• 

8.  The  focal  length /of  a  lens  is  given  by  the  formula 

---  +  - 

/        2h  ^  P-2  ' 


§  29.  SERIES    AS    A    MEANS   OF   COMPUTATION.  35 

where  Pi  and  p,  denote  two  conjugate  focal  distances.  Obtain  a 
simpler  approximate  formula  for  ./"  that  will  answer  when  p,  and  jh 
are  nearly  equal. 

•).  "A  ranchman  6  feet  7  inches  tall,  standing  on  a  level  plain, 
agrees  to  buy  at  S7  an  acre  all  the  land  in  sight.  How  much  nuist 
he  pay?  Given  640  acres  make  a  square  mile."  Admission  Exam, 
in  Sol.  Geom.,  June,  1895. 

Show  that  if  the  candidate  had  assumed  the  altitude  of  the  zone 
in  sight  to  be  equal  to  the  height  of  the  ranchman's  eyes  above  the 
ground  and  had  made  no  other  error  in  his  solution,  his  answer  would 
have  been  4  cents  too  small. 

10.  Show  that  for  small  values  of  h  the  following  equations  are 
approximately  correct  {h  may  be  either  positive  or  negative) 

(1  -J-  h)"'  =  1  -1-  mil  .    . 

Hence      {1  -\- hy  =  I  -\-  2Ji  ;  ^  1  -\-  h  =  1  -\-  ^h; 

1  .   _;  1_ 

1  +  /^-  ''  (1  +  /0' 

^  =  1  —  f^h  . 

V  1  -f  /i 

If  h,  Ic,  I,  p, are  all  numerically  small,  then,  approximately, 

(1  +/r)  (1  +  70  (1  +  /) =  1  +^'  +  ^-  +  /+ , 

(1  +  0  (i-hp) ^   ^ 


=  1-/';  7^-r-M2  =  i-2'^; 


III.    TAYLOR'S   THEOREM. 


30.  It  is  not  the  object  of  this  chapter  to  prove  Taylor's  Theorem, 
since  this  is  done  satisfactorily  in  any  good  treatise  on  the  Differ- 
ential Calculus  ;  but  to  indicate  its  bearing  on  the  subject  under  con- 
sideration and  to  point  out  a  few  of  its  most  important  applications. 

It  is  remarkable  that  this  fundamental  theorem  in  infinite  series 
admits  a  simple  and  rigorous  proof  of  an  entirely  elementary  nature. 
Rolle's  Theorem,  on  which  Taylor's  Theorem  depends,  and  the  Law 
of  the  Mean  lie  at  the  very  foundation  of  the  differential  calculus. 
From  Rolle's  Theorem  follows  at  once  the  theorem  contained  in  the 
equation 

fix,  +  70  =/(-^o)  +/'  (^o)  A  +/"  (xo)  ^  +  •  •  •  •  +r"  (a-o  +  eh)  ^' ,  (13) 

0  <  ^  <  1. 
This  latter  theorem  is  frequently  refen'ed  to  as   Taylor's   Theorem 
with  the  Remainder     i?„  =:  /<"'  (Xq  -\-  Oh)  —^     •     It  includes  the  Law 

of  the  Mean 

/(o-o  +  ^0  — ./'(-^o)  =  Kf  O^-o  +  Oh)  (14) 

as  a  special  case  and  thus  affords  a  proof  of  that  Law.  If  in  (13), 
when  n  increases  indefinitely,  R^  converges  towards  0  as  its  limit, 
the  series  on  the  right  hand  side  of  (13)  becomes  an  infinite  power 
series,  representing  the  function  fix,  -\-  h)  throughout  a  certain 
region  about  the  point  X(^ : 

fix,  +  h)=fix,)  J^fix,)h+r{x,)  1^  + (15) 

This  formula  is  known  as  Taylor's  Theorem  and  the  series  as  Taylor's 
Series. 

The  value  x,  is  an  arbitrary  value  of  x  which,  once  chosen,  is  held 
fast.  The  variable  x  is  then  written  as  x,  -\-  h.  The  object  of  this 
is  as  follows.  It  is  desired  to  obtain  a  simple  representation  of  the 
function /(x)  in  terms  of  known  elements,  for  the  purpose  of  com- 
puting the  value  of  the  function  or  studying  its  properties.  One  of 
the  simplest  of  such  forms  is  a  power  series  with  known  coefficients. 


§§80,31.  tavi.ok's  tiikouem.  37 

Now  it  is  usually  impossible  to  represent /(a;)  by  one  and  the  same 
power  series  for  all  values  of  a,',  and  oven  when  this  is  possible,  the 
series  will  not  converge  rapidly  enough  for  large  values  of  the  argu- 
meijt  to  1)6  of  use  in  computation.  Consequently  wo  confine  our 
attention  to  a  limited  domain  of  values,  choose  an  Xq  in  the  midst  of 
this  (loniaiii,  and  roplaco  the  independent  variable  x  by  A,  where 

X  =  x^,  -\-  h,  h  =  X Xq. 

The  vahies  of  .)•  for  the  domain  in  (luestion  may  not  be  small,  but  the 

vahies   of  //  will   l)c,  //  =  0  corresponding  to  x  =z  Xq.     If  x^  is  so 

chosen  that /(Xq), /'(a'o), /"(a'o), ad  in/,  are  all  finite,  then 

the  value  of  f(x)  for  values  of  x  near  to  .Tq,  i.  e.  for  values  of  h 

numerically  small,  will  usually*  be  given  by  Taylor's  Theorem. 

An  example  will  aid  in  making  clear  the  above  general  statements. 

Let 

f(x)  =  log  X. 

Then  it  is  at  once  clear  that  f(x)  cannot  be  developed  l)y  Taylor's 
Theorem  for  Xq  =z  0,  for/(0)  =  log  0  :=  —  x  .  It  is  just  at  this 
point  that  the  freedom  that  we  have  in  the  clioice  of  .fy  stands  us  in 
good  stead;   for  if  we  take  x^  greater  than  0,  then  /(a-'o),  /' (x^), 

/"(Xq), will  all  be  finite  and /(.ro -|- /i)  can  be  developed  by 

Taylor's  Theorem,  the  series  converging  for  all  values  of  h  lying 
between  x^  and  —  Xq.  The  proof  is  given  for  Xq  =:  1  in  the  Dijf'. 
CuL,  §  130.  Thus  we  have  a  second  proof  of  the  development  of 
log  (1  +  h),  (formula  (8)  of  §  19). 

31.  Ttco  Applications  of  Tuijlofs  Theorem  with  the  Remainder, 
(13).  This  theorem,  it  will  i)e  observed,  is  not  a  theorem  in  infinite 
series.  Any  function  whose  first  n  derivatives  are  continuous  can  l)e 
expressed  in  the  form  (13),  while  the  expression  in  the  form  (15) 
requires  the  proof  of  the  possibility  of  passing  to  the  limit  when 

?l  zz=    X  . 

Thus  (13)  is  a  more  general  theorem  than  (15)  and  it  avoids  the 
necessity  of  a  proof  of  convergence.!  It  is  because  of  the  applica- 
tions that  (13)  and  (15)  have  in  common,  that  it  seemed  desirable  to 
treat  some  applications  of  (13)  here.  * 

*  Exceptions  to  this  rule,  though  possible,  are  extremely  rare  in  ordinary 
practice. 

t  It  is  desirable  that  (13)  should  be  applied  much  more  freely  th;in  has 
hitiierto  been  the  custom  in  works  on  the  Infinitesimal  Calculus,  both  bi'cau<e 
it  att'onls  a  simple  means  of  proof  in  a  vast  variety  of  cases  and  because  many 
proofs  usually  given  by  the  aid  of  (15)  can  be  simplified  or  rendered  rigorous 
by  the  aid  of  (13).     The  applications  given  in  this  section  are  cases  in  point. 


38  Taylor's  theorem.  §  31. 

First  Aj^plication :  Maxima,  Minima  and  Points  of  Inflection; 
Curvature.  Let  it  be  required  to  study  the  function  f(x)  in  the 
neighborhood  of  the  point  x  ^z  x^. 

f(Xo  +  ^0  =  /(-^'o)  +  /'  (•■^•o)  h  +  ^  /"  (a-o  +  eh)  h\ 
Plot  the  function  as  a  cui*ve  :  * 

.2/1  =/(•'•)  =/(-«o  +  h)^ 
and  plot  the  cui've 

The  latter  curve  is  a  right  line.  Consider  the  difference  of  the  ordi- 
nates,  ?/i  and  ?/., : 

Hence  it  appears  that  y^  —  y^  is  an  infinitesimal  of  the  second  order. 
This  property  characterizes  the  line  in  question  as  the  tangent  to  the 
curve  in  the  point  .^o,  and  thus  we  get  a  new  proof  that  the  equation 
of  the  tangent  is 

y  =  f(xo)  -\-  f'(xo)  (x Xo). 

Next,  suppose 

/'(xo)  =  0,     r"'-'H^o)  =  0,      /^^"H^o)  >  0. 

Then  f(x,  +  h)  =  f(x,)  +  /(^")  (x,  +  Oh)  ^^  • 

The  equation  of  the  tangent  is  now 

2/2  =  f(Xo) 
and  2/1  —  2/2  =  f'"'  (^o  +  Oh)  ^|^ 

f-^"^  (x)  vn.\\  in  general  be  continuous  near  the  point  x  z=  Xq  and 
it  is  positive  at  this  point ;  it  will  therefore  be  positive  in  the 
neighborhood  of  this  point  and  hence 

2/i  —  2/2  >  0 

both  for  positive  and  for  negative  values  of  h,  i.  e.  the  curve  lies 
above  its  tangent  and  has  therefore  a  minimum  at  the  point  x  =  Xq. 

Similarly  it  can  be  shown  that  if  /(^'O  (^o)  <  0,  all  the  earlier  deri- 
vatives vanishing,  f(x)  has  a  maximum  in  the  point  a'o. 


Lastly,  let 

f'(Xo)*=  0,      P'-H^'o)  =  0,       /^^"^"(^•o)  H=  0. 

*  The  student  should  illustrate  each  case  in  tliis  §  by  a  figure. 


§31.  Taylor's  THEOREM.  39 

Then      y,  —  y,  =  f^"  +  '^  (./•„  +  6h)         ^'"'^' 


(2n-\-  1)! 

y(2,.  +  i)^^-j  Avill  ill  general  he  conliiiuoiis  near  x  :=  Xq  and  it  will 
therefore  preserve  the  same  sign  for  small  values  of  /i,  positive  or 
negative;  but  /i^"  +  ^  changes  sign  witii  //.  Hence  the  curve  lies  on 
opposite  sides  of  its  tangent  on  opposite  sides  of  the  point  x^  and 
this  is  then  a  point  of  inflection. 

Exercises.  1 .  Show  that  the  condition  for  a  point  of  inflection  not 
parallel  to  the  a'-axis  is 

/"(.^•o)  =:  0,    r'"^(x,)  z=  0,     r'"^'^(xo)  dp  0, 

/  2«  +  i)  ^^^  being  continuous  iieur  r  =  Xq. 

2.  Show  that  a  perpendicular  drawn  to  the  tangent  from  a  point 
P'  infinitely  near  to  a  point  of  inflection  P  is  an  infinitesimal  of 
higher  order  than  the  second. 

Curvature.  The  osculating  circle  was  defined  (Diff.  Cal.  §  90)  as 
a  circle  tangent  to  the  given  curve  at  P  and  having  its  centre  on  the 
inner  normal  at  a  distance  p  (the  radius  of  curvature)  from  P.  We 
will  now  show  that  if  a  point  P'  be  taken  infinitely  near  to  P  and  a 
perpendicular  P'M  be  dropped  from  P  on  the  tangent  at  P,  cutting 
the  osculating  circle  at  P",  then  P'P"  is  in  general  an  infinitesimal 
of  the  third  order  referred  to  the  arc  PP  as  principal  infinitesimal. 
Let  P  be  taken  as  the  origin  of  coordinates,  the  tangent  at  P  being 
the  axis  of  x  and  the  inner  normal  the  axis  of  // ;  and  let  the  ordinate 
y  be  represented  by  the  aid  of  (13).     Here 

a-o=0,         x=h,        f(0)=f\0)=0,        /"(0)>0, 

and  y  =  ^f"  (0)  .r^  +  }/'"  (Ox)  x'^. 

The  radius  of  curvature  at  P  is 


a 


p  - 


DJ'y  /"(O) 

and  the  equation  of  the  osculating  circle  is 

^'''  +  (.'/  —  Pf  =  P- 
Hence  the  lesser  ordinate  y'  of  this  circle  is  given  by  the  formula 


X* 


y'  =  p  -  ^  p^  -  x^  =  p  -  p  (1  -  I  --  l-  -^^ ) 

*  Instead  of  the  infinite  series,  formula  (13)  might  have  been  used  here,  with 
n  =  4.  But  we  happen  to  know  in  tliis  case  that  the  function  can  be  developed 
by  Taylor's  Tlieorcni  (15).  .. 


40  TAYLOR'S    THEOREM.  §§  31,  32. 

and         y-y'^  -c'  (^if"'{ex)  -  i  i  -'^ )  • 

From  this  result  follows  that  (y  —  y')/x^  approaches  in  general 
a  finite  limit  different  from  0,  and  hence  that  y  —  y'  is  an  infini- 
tesimal of  the  third  order,  referred  to  P'M  =  a;  as  principal  infini- 
tesimal. But  P'M  and  PP'  are  of  the  same  order.  Hence  the 
proposition. 

Exercise.  Show  that  for  any  other  tangent  circle  y  —  y'  is  an 
infinitesimal  of  the  second  order. 

Second  Application :  Error  of  Observation.  Let  x  denote  the  magni- 
tude to  be  observed,  ?/  =  /  (x)  the  magnitude  to  be  computed  from 
the  observation.  Then  if  .Jo  be  the  true  value  of  the  obseived  magni- 
tude, X  ^  Xq  -\-  h  the  value  determined  by  the  observation,  h  will  be 
the  error  in  the  observation,  and  the  error  H  caused  thereby  in  the 
result  will  be  (c/,  (14)) 

H  =  fix,  +  h)  —  /(a-o)  =  /'  (X,  +  Oh)  h. 

In  general  f'(x),  will  be  a  continuous  function  of  x  and  thus  the 
value  of  f(xo  -\-  6h)  will  be  but  slightly  changed  if  x^  -^  Oh  is 
replaced  by  x.     Hence,  approximately, 

H  =  f'{x)h 

and  this  is  the  formula  that  gives  the  error  in  the  result  due  to  the 
error  in  the  observation. 

32.  The  Principal  Apx>lications  of  Taylor's  Theorem  icithout  the 
Remainder,  i.  e.  Taylor's  Series  (15)  consist  in  showing  that  the 
fundamental  elementary  functions:  e"^,  sina:;,  cos.r,  log.i;,  x^,  sin~^.^*, 
tan~^.c  can  be  represented  by  a  Taylor's  Series,  and  in  determining 
explicitly  the  coefficients  in  these  series.  It  is  shown  in  Ch.  IX  of 
the  Diff.  Col.  that  these  developments  are  as  follows.* 


e'-" 


x^     .     x° 


1  +  -^^  +  9T  +  sT  + 


^.3  ^.5 


sinxr.  ...--+_, 


X^      ,       X* 


COS  .T  =  1  —  -  +  4  , 
These  developments  hold  for  all  values  of  x. 

*  The  developments  for  sin~'x  and  tan- 'a;  are  to  be  sure  obtained  by  in- 
tegration ;  but  the  student  will  have  no  difficulty  in  obtaining  them  directly 
from  Taylor's  Theorem. 


§§  32,  33.                    Taylor's  theorem.  41 

loga;  =\og(l  -\-  h)=  h  —  ~  -\-  ^— 

x^^  =  (!-{-  hy  z..  1  +  ;x/.  +  '"\~^^^  /^^  + 

'    2  3     '    2  •  4    .5     ' 

x^    ,    x-^ 
tan-i.t;  =  .x  —  -  +  -  — 

These  developments  hold  for  all  values  of  h  (or,  in  the  case  of  the 
last  two  formulas,  of  x)  numerically  less  than  1. 

Exercise.  Show  that  sin  a;  can  be  developed  aliout  anj'  point  x,,  by 
Taylor's  Theorem  and  that  the  series  will  converge  for  all  values  of  h. 
Hence  comi)ute  sin  46°  correct  to  eight  places  of  decimals. 

33.  As  soon  however  as  we  pass  beyond  the  simple  functions  and 
try  to  apply  Taylor's  Theorem,  we  encounter  a  ditliculty  that  is 
usually  insurmountable.  In  order  namely  to  show  that  f{x)  can  be 
expanded  by  Taylor's  Theorem  it  is  necessary  to  investigate  the 
general  expression  for  the  n-i\\  derivative,  and  this  expression  is 
usually  extremely  complicated.  To  avoid  this  difHculty  recourse 
is  had  to  more  or  less  indirect  methods  of  obtaining  the  expansion. 
For  example,  let  it  be  required  to  evaluate 


X 


1  ef  —  e~''  , 

■ ax. 

0  X 


The  indefinite  integral  cannot  be  obtained  and  thus  we  are  driven  to 
develop  the  integrand  into  a  series  and  integrate  term  by  term.  Now 
if  we  try  to  apply  Taylor's  Tlieorem  to  the  function  {f  —  e~'')/x, 
the  successive  derivatives  soon  become  complicated.  We  can  how- 
ever proceed  as  follows : 

X-         x^ 
e^  =  1  +  .r  +  .,  ,  -f  ..  ,  + , 


o—x 


X^  X^ 


=  '-^+2-!-3!  + 


e-     —  .       ..    ,    ^^ 


/          .)■«        x^  \ 

e- _, -X  =  2  (^.r  4-  _.,,  +  _.,+ j; 

and  hence,  dividing  through  by  x,  wo  have 


42  TAYLOR'S    THEOREM.  §  33. 

X^^'"-^=K'+3^  +  ^+ )  =  2.114    502. 

Examples.     Do  the  examples  on  p.  50  of  the  Problems. 

General  Method  for  the  Expansion  of  a  Function.  To  develop  a 
function  /(•'c),  made  up  in  a  simple  manner  out  of  the  elementary 
functions,  into  a  power  series,  the  general  method  is  the  following. 
The  fundamental  elementary  functions  having  been  developed  by 
Taylor's  Theorem,  §  32,  we  proceed  to  study  some  of  the  simplest 
operations  that  can  be  performed  on  series  and  thus,  starting  with 
the  developments  already  obtained,  pass  to  the  developments  de- 
sired. 


IV.    ALGEBRAIC    TRANSFORMATION'S 

OF    SERIES. 


34.  It  has  been  pointed  out  repeatedly  (§§19,  21,  24)  that  since 
an  infinite  series  is  not  a  sum,  ])nt  a  limit  of  a  sum,  processes  appli- 
cable to  a  sum  need  not  be  applicable  to  a  series ;  if  applicable,  this 
fact  requires  proof. 

For  exanii)le,  the  value  of  a  sum  is  independent  of  the  order  in 
which  the  terms  are  added.  Can  this  interchange  in  the  order  of  the 
terms  be  extended  to  series?     Let  us  see.     Take  the  series 

1  -  i  +  ^-  -  i  + 
Its  value   is    less   than       1  —  A  -)-  a  z 
terms  as  follows  : 

1  +  i  -  i  H-  i  +  I  -  i  +  i  -f 

The  general  formula  for  throe  successive  terms  is 


(«) 

--  t      (§  12) 

Rearrange   its 

1         I 

111          CI 

(/?) 

4A;  — 3    '    4fc— 1        2A; 
and  if  each  pair  of  positive  terms  be  enclosed  in  parentheses : 

(i  +  :U-.^  +  (.^  +  4)-i  +  (^  +  TV)-i+ (y) 

the  result  is  an  alternating  series  of  tlie  kind  considered  in  (§11). 
For  it  is  easy  to  verify  the  inequalities 

Hence  the  series  (y)  converges  toward  a  value  greater  than  (1  -|-  |^) 

—  A  =  f .  The  sum  of  the  first  n  terms  of  (^)  differs  from  a  properly 
chosen  sum  of  terms  of  (y)  at  most  by  the  first  term  of  a  parenthesis, 

—  a  quantity  that  ai)proaches  0  as  its  limit  when  »  ^  x  .  lleuee 
the  series  (/3)  and  (y)  have  the  same  value  and  the  rearrangement  of 
terms  in  (a)  luis  thus  led  to  a  series  (/3)  having  a  different  value 
from  (a). 

In  fact  it  is  possible  to  rearrange  the  terms  in  (a)  so  that  tlie  new 
series  will  have  an  arbitrarily  preassigned  value,  C.  For,  if  C  is 
positive,  say  10  000,  liegin  by  adding  from  the  positive  terms 

1  +  ^  +  ^H- 


44  ALGEBRAIC    TRANSFORMATIONS    OF    SERIES.      §§  34,  35. 

till  enough  have  been  taken  so  that  their  sum  will  just  exceed  C. 
This  will  always  be  possible,  since  this  series  of  positive  terms 
diverges.     Then  begin  with  the  negative  terms 


1 1 1 

9  4 


4  E" 

and  add  just  enough  to  reduce  the  sum  below  C.  As  soon  as  this 
has  been  done,  begin  again  with  the  positive  terms  and  add  just 
enough  to  bring  the  sum  above  C;  and  so  on.  The  series  thus 
obtained  is  the  result  of  a  rearrangement  of  the  terms  of  (a)  and  its 
value  is  C. 

In  the  same  way  it  can  be  shown  generally  that  if 

?/0   +    «1    +    «2   + 

is  any  convergent  series  that  is  not  absolutely  convergent,  its  terms 
can  be  so  rearranged  that  the  new  series  will  converge  toward  the  pre- 
assigned  value  C.  Because  of  this  fact  such  series  are  often  called 
conditionally  convergent,  Theorem  1  of  §  35  justifying  the  denoting 
of  absolutely  convergent  series  as  uncoyiditionally  convergent. 

There  is  nothing  paradoxical  in  this  fact,  if  a  correct  view  of  the 
nature  of  an  infinite  series  is  entertained.  For  a  rearrangement  of 
terms  means  a  replacement  of  the  original  variable  s^  by  a  new  vari- 
able .s',^ ,  in  general  unequal  to  s^ ,  and  there  is  no  a  x>Tiori  reason  why 
these  two  variables  should  approach  the  same  limit. 

The  above  example  illustrates  the  impossibility  of  extending  a 
priori  to  infinite  series  processes  applicable  to  sums.  Most  of  such 
processes  are  however  capable  of  such  Q-s.ie.nB\o\\  under  ^rroper  restric- 
tions, and  it  is  the  object  of  this  chapter  to  study  such  extension  for 
some  of  the  most  fundamental  processes. 

35.    Theorem  1.     In  an  absolutely  convergent  series  the  terms  can 
be  rearranged  at  j:)leasure  without  altering  the  value  of  the  series. 
First,  suppose  all  the  terms  to  be  positive  and  let 

Sn  =  '^0 -{-  ^h -\- +  w„-i ;  lim   s„  =  U. 

n  =  CO 

After  the  rearrangement  let 

S'„'  =  <  +  u\  -\- +  u\^,_i . 

Then  s'„.  approaches  the  limit  U  when  n'  =  co  .  For  s\^.  always  in- 
creases as  n'  increases  ;  but  no  matter  how  large  n'  be  taken  (and  then 
hold  fast),  n  can  (subsequently)  be  taken  so  large  that  s^^  will  include 
all  the  terms  of  s\^'  and  more  too ;  therefore 


§  35.  ALGEBRAIC    TRANSFORM ATIOXS    OF    SERIES.  45 

or,  no  matter  liuw  hirgu  a'  be  taken, 

S'n'    <    U' 

Hence  .s'^^,  approaches  a  limit  U'  <C  U. 

"We  may  now  turn  thintrs  al)oiit  and  regard  the  w-series  as  gener- 
ated by  a  rearrangement  of  the  teiiiis  of  the  w'-series,  and  the  above 

reasoning  shows  that  U  ^  U'  \  hence  U'  =  U.  q.  e.  d. 

Exercise.  The  second  step  in  the  above  proof  was  abbreviated 
by  an  ingenious  device.  Replace  this  dcNice  by  a  direct  line  of 
reasoning. 

Secondly,  let  the  series 

«o  +  '<i  +  ''2  H- 

be  any  absolutely  convergent  series  and  let 

■^,  =  cr,,,  -  r„  ,  (Cf.  §  14) 

U=V—  w. 

Let  ii'q  -\-  v\  -|-  "'2 

be  the  series  after  the  rearrangement  and  let 

U'  =  V  —  w . 

But  V  =  V  and  W  =  W;  hence  U'  —  U. 
Exercise.     Find  the  value  of  the  series 

l-Ll_l-Ll_J_l_ill.jL_l_L    . 
91    I     08         '2'^         •>5    '     2''         9*         99    '     '2^^         9*    ' 


•        •       • 


Theorem  2.     Jf 

U=  ?/o  +  "i  + 

V=v,-^v,-\- 

are  any  two  convergent  series,  tlieij  can  he  added  term  by  term,  or 

U^  V=  vo  +  r,  +  »,  +  ,.,  +  ..,+ 

If  they  are  absolutely  convergent,  the  third  series  ivill  also  be  abso- 
lutely  convergent  and  hence  its  terms  can  be  rearranged  at  pleasure. 
Let 

^.  =  "o  -h  "i  + 4-  ",.-1. 

Then 

•*'„  +    fn   =   («0   +    ^-o)  +  ("1    -f    ^'0  4- +    (»„-!    4-    i'„-l). 


46  ALGEBRAIC    TRANSFORMATIONS    OF    SERIES.  §  35. 

When  n  =z  CO  ,  the  left  baud  side  converges  toward  U  -{-  V\  hence 

U+  V=  («o  +  ^'o)  +  (^h  +  t'l)  + 

It  remains  to  show  that  the  parentheses  may  be  dropped.  This  is 
shown  in  the  same  way  as  in  the  case  which  arose  in  §  34. 

The  proof  of  the  second  part  of  the  theorem  presents  no  difficulty 
and  may  be  left  to  the  student. 

Exercise.     Show  that  if 

U  =  "o  +  wi  +  "2  + 

is  any  convergent  series,  c  any  number, 

CU  =^  CUq  -^  CUi  -{-  c  n^  -\- 

Theorem  S.     If 

U  =  »„  +  "1  -f  ?'2  + 

y  =  ''o  -r  '"i  +  '"2  -r 

are  any  two  absolutely  convergent  series,  they  can  he  multiplied  together 
like  sums;  i.  e.  if  each  term  in  the  Jirst  series  be  multiplied  into  each 
term  in  the  second  and  the  series  of  these  2^>'oducts  formed,  this  series 
will  converge  absolutely  toward  the  limit  UV.     For  example 

UV=    UqVq   -\-    «o^'l    +    "I'V   +    "o'"2    +    »l''l    +    ''2'-"0   + 

This  theorem  does  not  in  general  hold  for  series  that  are  not 
absolutely  convergent. 

Let  .s„  =  ?/o  H-  »i  + +  ^/„_i , 

K  =  ''0  +  ''i  + +  ^„-i ; 

then  liin    .s  t    =  UV. 

n    li 
W   =  00 

The  terms  of  the  product  s^t^^  are  advantageously  displayed  in  the 
following  scheme.  They  are  those  terms  contained  in  a  square  n 
terms  on  a  side,  cut  out  of  the  upper  left  hand  corner  of  the  scheme. 

-••■■'     i  .-•''      '  ••'■'^     I  .-;■'' 
/""^         ,■-■■'  .--'      I      .    '     I    ~ 

.•■"  I  -•■'  ' 


The  theorem  asserts  that  if  any  series  be  formed  by  adding  the 
terms  of  this   scheme,  each  term  appearing  in  this  series  once  and 


§  35.  ALGKHUAIC    TKANSFOIiMATIONS    OV    SRKIES.  47 

only  once,  —  for  example,  the  terms  that  lie  on  the  oblique  lines,  the 
successive  lines  being  followed  fiom  top  to  Ijottom  : 

Wo^'o  +  "o^'l  +  "i^'u  +  "o''2  + ,  (a) 

this  series  will  converge  absolutely  toward  the  limit  UV. 

It  is  siitllcient  to  show  that  one  series  formed  in  the  prescribed  way 
from  the  terms  of  the  scheme,  for  example  the  series  formed  by  fol- 
lowing the  successive  boundaries  of  the  squares  from  top  to  bottom 
and  then  from  right  to  left,  namely  the  series 

converges  absolutely  toward  the  limit  UV.  For  any  other  series 
can  then  be  generated  by  a  rearrangement  of  the  terms  of  this  series. 

Let  S_y  denote  the  sum  of  the  tirst  N  terms  in  (/3). 

First  suppose  all  the  terms  of  the  /(-series  and  the  v-series  to  be 

positive.*     Then,  if  n^  <  iVr<  (n  +  1)^, 

^nK     S     ^■''     S     ^''  +  l^n  +  l- 

Hence  lim    8^=  UV. 

N=  00 

Secondly,  if  the  w-series  and  the  v-series  are  any  absolutely  con- 
vergent series,  form  the  series  of  absolute  values 

«'o  +   «'.  +   "'.  + 

v\  +  v\  +  c',  -h 

The  product  of  these  series  is  the  convergent  series 

w>''o  +  "'o*"'i  4-  "'it"'u  +  "'o«''2  + 

But  this  series  is  precisely  the  series  of  absolute  values  of  (a),  and 
therefore  (a)  converges  absolutely.  It  remains  to  show  that  the 
value  toward  which  it  converges  is  UV.  Since  Sy  approaches  a 
limit  when  jV,  increasing,  passes  through  all  integral  values,  S^  will 
continue  to  approach  a  limit,  and  tiiis  will  l)e  the  same  limit,  if  ^^ 
passes  only  through  the  values  }r  : 

lim    Sy  =  lim    S„2 . 
JV  =  CO  n  =  00 

But 

S„,  =  .-?,.^.  and  lim    S„,  =  UV. 

n  =  x> 
This  proves  the  theorem. 

♦  The  case  that  some  of  the  terms  are  0  must  not  however  be  exchided  ;  hence 
the  double   sign    ( ^ )   in   the    inequality    below:    Sy  "^  s„  +  \t„  +  \. 


48  ALGEBRAIC    TRANSFORMATIONS    OF    SERIES.        §§35,36. 

For  example,  let 

f{x)  =  tto  +  a^x  -\-  a^x"^  -f , 

^(x)  =  &o  +  ^1-^'  +  ^2^^  + 

be  two  convergent  power  series,  x  any  point  lying  at  once  within  the 
region  of  convergence  of  both  series.  Then  the  product  of  these 
series  is  given  by  the  formula 

This  formula  can  be  used  to  give  the  square,  or  by  repeated  appli- 
cation, any  power  of  a  power  series.  Thus  it  gives  as  the  square  of 
the  geometric  series 

1  +  •-«  +  ^^  + 

the  series 

1  4-  2  X-  -f-  3  x'2  -(-   .   .   .   .   .   , 

a  result  agreeing  with  the  binomial  expansion  of  (1  -j-  x)~^. 
Exercise.     Find  the  first  four  terms  in  the  expansion  of 

X  ,  log  (\  -\-  x') 

and  »v     -r     y 


V  1  —  .^■'  1  +  ^ 

Square  the  series  for  e-^  and  show  that  the  result  agrees  ^ith  the 
expansion  of  e^. 

36.    One  more  theorem  is  extremely  useful  in  practice.     Its  proof 
would  carry  us  beyond  the  bounds  of  this  chapter. 
Let 

^niy)  =  &0  +  Wy  +  hy'  + +  Ky" 

be  any  polynomial  in  y  and  let  y  be  given  by  the  convergent  power 
series  in  x  : 

Then  the  powers  of  y  :  ?/'^,  y^, y"  can  be  obtained  at  once  as 

power  series  in  x  by  repeated  multiplications  of  the  .r-series  by  itself, 
the  terms  of  the  polynomial  <f>^^  (y)  then  formed  by  multiplying  these 
power  series  respectively  by  the  cofficients  6,  and  the  polynomial 
<f>^^  (y)  thus  represented  as  a  power  series  in  x  by  the  addition  of  these 
terms. 

Suppose  however  that  instead  of  the  polynomial  <}>^  (y)  we  had  an 
infinite  series  : 

<l>  {y)  =  h-\-  Ky  +  ^2y''  + 

Under  what  restrictions  can  the  above  process  of  representing 
<f)^^  (y)  as  a  power  series  in  x  be  extended  to  representing  <^  (y)  as  a 
power  series  in  cc? 


§  3().  ALGEHRAIC    TRANSFORMATIONS    OF    SERIES.  49 

One  restriction  is  imniediutely  obvious.  Since  a  power  series 
represents  a  continuous  function  (y.  §  38)  the  values  of  y  corre- 
sponding to  small  values  of  x  will  lie  near  to  cIq  and  thus  the  2:)oint  % 
must  sureli/  lie  tvithin  the  interval  of  convergence  ( —  t  <^y  <^  r)  of 
the  series  cf>(y).  Suppose  a^  =  0  \  then  this  condition  is  always 
satisfied.  And  now  our  theorem  is  precisely  this,  that  no  further 
condition  is  necessary. 

Theorem  4.  If  «o  =  0»  '"-'  f  ether  restriction  is  necessary;  i.e. 
the  above  process  of  rejjresenting  <f>  (y)  as  a  jyoiver  series  in  x  is  alicays 
applicable.* 

Remark.  The  point  of  the  theorem  just  quoted  is  this.  We  know 
from  §  35  that  each  term  in  the  y  series  can  be  expressed  as  a  power 
series  in  x : 

b„y"  =fix)  z=  ao^")  +  a,^"^x  +  a^C'^x''  + 

and  hence  that  <f>  (y)  can  be  expressed  in  the  form 

^(^y)=f^(x)-\-f{X)+f,ix)-j- 

It  remains  to  prove  (and  it  is  precisely  this  fact  that  the  theorem  as- 
serts, —  a  fact  not  true  in  general  of  a  convergent  series  of  the  form 

fo{x)-{-A{x)+f{x)-\- , 

where  f„(x)  denotes  a  power  series)  that  if  we  collect  from  these 
series  all  the  terms  of  common  degi-ee  in  x  and  then  rearrange  them 
in  the  form  of  a  single  power  series,  first,  this  series  will  converge, 
and  secondl3%  its  value  will  l»e  <^ (,'/)• 

Examples.  1.  Let  it  be  required  to  develop  e*»'"^  according  to 
powers  of  x.t     Let  y  =  x  sin.r.     Then 

cl^U,)  =  e"  =  1  +  ,/  +  hir  +  i/  4-  iii/'  4- 

,,  ,.2  1    )•■!  _1_       1      1-6  1        r*  -4- 

il/'=  ^^'     —  tV^'+ 

^y'=  ^^'-^ 

♦  The  case  n„  =  0  is  the  oni'  tliat  usually  arises  in  practice.  But  the  theorem 
still  holds  provided  only  that  —  r  <^  a„  ■<  r,  the  only  difference  being  that  the 
coettifionts  in  the  final  scries  will  then  be  infinite  series  instead  of  sums.  Cf. 
Stolz.  Allgemeine  Aritlimfiik.  Vol.  I,  Cli.  X,  §25. 

t  Even  wlien  it  is  known  tliat  a  function  can  be  developeii  by  Taylor's 
Theorem  (v.  Ch  111:  Diff.  Cal.,  Ch.  IX;  Ini.  Cat..  Ch.  XVin  it  is  usually 
simpler  to  dt'ltrinini'  the  coefficients  in  the  series  by  the  method  hero  set  forth 
than  l)y  performing  the  successive  differentiations  requisite  in  the  application  of 
Taylor's  formula.     The  example  iu  liand  illustrates  the  truth  of  this  statement. 


50  ALGEBRAIC    TRANSFORMATIONS    OF    SERIES.  §  36. 

2.  Find  the  first  4  tei-ms  in  the  expansion  of  sin  (Zc  sin  a;). 

3 .  Obtain  a  few  terms  in  the  development  of  each  of  the  following 
functions  according  to  powers  of  x. 

log  cos  a;. 

Suggestion.     Let  cos  a-  =z  I  -\-  y-  then 

y  =  —  i  -^"^  +  2T  •'^■^  — 

and  log  cos x  =  ^  ^x^  —  t^^*  —  4T -*^^  "l~ 


1  1 


V  1  —  2.t;  cos  ^  -t-  a;2  ^  i  _  A:2  sin2a; 

Theorem  4  gives  no  exact  information  concerning  the  extent  of 
the  region  of  convergence  of  the  final  series.  It  merely  asserts  that 
there  is  such  a  region.  This  deficiency  is  supplied  by  an  elementary 
theorem  in  the  Theory  of  Functions.* 

But  for  many  applications  it  is  not  necessary  to  know  the  exact 
region  of  convergence.  For  example,  let  it  be  required  to  determine 
the  following  limit. 

log  COSX  -j-    1 =^===: 

lim     V  1  +  ^^  +  '^ 

x  =  0  sin.x  —  X 

Both  numerator  and  denominator  can  be  developed   according  to 
powers  of  x.     The  fraction  then  takes  on  the  form 

^  x^  -\-  higher  powers  of  x 


—  ^x^  -\-  higher  powers  of  x 

Cancel  x^  from  numerator  and  denominator  and  then  let  x  approach 
0  as  its  limit.  The  hmit  of  the  fraction  is  then  seen  to  be  —  3.  The 
usual  method  for  dealing  with  the  limit  0/0  is  applicable  here,  but 
the  method  of  series  gives  a  briefer  solution,  as  the  student  can 
readily  verify. 

Example.     Determine  the  limit 


lim     s/  a^  —  ^^  —  V  «^  +  ^'^ 
x  =  0  1  —  cosa; 

An  important  application  of  Theorem  4  is  to  the  proof  of  the 
following  theorem. 

*  Cf.  Int.  CaL,  §220;   Higher  Mathematics,  Ch.VI,  Functions  of  a  Complex 
Variable,  by  Thos.  S.  Fiske ;  John  Wiley  &  Sons. 


§  3().  ALGEBRAIC    TKANSFOKMATIOXS    OF    SEHFES.  51 

Theorem.      The  (jnotient  of  two  power  series  can  he  represented 

as  a  poicer  seiies,  jn-oridcd  the  constant  term  in  the  denominator  series 
is  not  0  : 

60  +  b^x.  -\-  boX-  -f _  ^ 

j j o — i —   (^0  ~\~  Cl^  ~\~  C2X    -4- , 

fto  +  "i-c  +  Wo'*-    + 

if  fto  -|=  0. 

It  is  sufliciciit  to  show  that 

1 


can  lie  so  represented,  for  tlien  the  power  series  that  represents  it 
can  be  multiplied  into  the  numerator  seiies 

Let  y  =  a^x  -\-  a.^x"^  -{- 

1        _1  1       _l_J/__i_l^_^_l_ 


provided  y/uo  is  numerically  less  than  unity,  i.  e.  y  numerically  less 
than  a^).  Thus  the  conditions  of  Theorem  4  are  fulfilled  and  the  func- 
tion l/(ao  -|-  y)  can  be  expressed  as  a  power  series  in  x  by  develop- 
ing each  term  ( —  1)"  2/"A^, +  1  i'^to  such  a  series  and  collecting  from 
these  series  the  terms  of  like  degree  in  x. 

CoROLLARV.     //'  the  coefficients  of  the  Jirst  m  poicers  of  x  in  the 
denominator  series  vanish,  the  quotient  can  be  expressed  in  the  form 

/...  +  i^ .,•  +  i, .r^  + _  C'_       (7-,„^x        4.  ^  _U 


a  0(^4- a    .10;"'  +  ^ -h iC"     '      .T"'-i     '  '      X 

For 

b^ -\- b^x -\- b^x^ -\- _  1     61  -(-  ^1  •^'  +  ^2'^'2  + 


1    (Co-[-Cx.i-+o,.r+ ) 


X" 

and  it  only  remains  to  set  c^  =1  ^„-m  ^"^  divide  x"'  into  each  term. 
Examples.     Show  that 

1              2 
tana;  =  x  -t-  ^  .i-^  +  ^  .r^  + , 

ctn.r  =z .1-  —  T^  .t    + 

.r        3  4o 

and  develop  sec  x  and  esc  x  to  three  terms. 

A  more  convenient  mode  of  determining  the  coetlicients  in  these 
expansions  will  be  given  in  §  38 . 


y.  conti:n^uity,  integration  and 

DIFFERENTIATION  OF  SERIES. 


37.  Continuity.  We  have  had  numerous  examples  in  the  fore- 
going of  continuous  functions  represented  by  power  series.  Is  the 
converse  true,  namely,  that  every  power  series  represents,  within  its 
interval  of  convergence,  a  continuous  function?  That  this  question 
is  by  no  means  trivial  is  shown  by  the  fact  that  while  the  continuous 
functions  of  ordinary  analysis  can  be  represented  (within  certain 
limits)  by  trigonometric  series,  i.  e.  b}^  series  of  the  form 

(/o  -|-  «i  cos  X  -\-  a^  cos  2  a;  -|- 

-|-  &iSina;  -f-  ^2  sin  2  cc  -|- 

a  trigonometric  series  does  not  necessarily,  conversely,  represent  a 
continuous  function  throughout  its  interval  of  convergence. 

Let  us  first  put  into  precise  form  what  is  meant  by  a  continuous 
function.     <^(a:)  is  said  to  be  continuous  at  the  point  x^  if 

lim     ^  {x)  z=.  (J3  (Xq)  ; 

i.  e.  if,  a  belt  being  marked  off  bounded  by  the  hues  y  =  (I>(xq)  -|-  e 
and  y  =^  <fi  (xq)  —  e,  where  e  is  an  arbitrarily  small  positive  quantity, 


Fig.  8. 


an  interval  (Cj  —  8,  ;«,,  4~  ^)»  ^  ^  0,  can  then  ahva3^s  be  found  such 
that,  when  x  lies  within  this  interval,  <j>(x)  will  lie  within  this  belt. 
These  conditions  can  be  expressed  in  the  following  form : 


§  37.  CONTINUITY,    INTEGRATION,    Diri'KKENTIATION.  53 

or  *  I  <^ (x)  —  «^ (a-o)  I  <  c ,  \  x  —  Xo\  <  8. 

A  simple  sulKcient  condition  that  the  series  of  continuous  functions 

?<„  (x)  -\-  7ii{x)-\- 

represent  :i  contimious  function  is  given  by  the  following  theorem. 
Theorkm  1.     If 

"o  (x)  -h  iiii^)  -{- •.  a  <  X  <  /8  , 

is  a  .series  of  cuntiniious  functions  couceryent  throuyhout  the  interval 
(a,  /?),  then  the  function  f  (x)  represented  by  this  series  will  be  con- 
tinuous throuyhout  this  inferral,  if  a  set  of  positive  numbers,  Mq,  J/^, 
M2, ,  independent  of  x,  can  be  found  such  that 

1)  I  a,^(x)   I  <  3/  ,     a<x<(3,      n  =  0,1,2, ; 

2)  ^A.  +  3/,  -\-M,-^ 

is  a  converyent  series. 

We  have  to  show  that,  Xq  being  any  point  of  the  interval,  if  a  posi- 
tive quantity  c  be  chosen  at  pleasure,  then  a  second  positive  quantity 
8  can  be  so  determined  that 

I  /(•^•)  —/(•'•„)   I  <  e  ,  if  I  a-  —  Xo  |<  S  . 

Let 

^s„(^0=  "0  (■'•)+  "i(-0+ -\-u^-i{x), 

f(x)  =  s,^{x)^r,^{x). 
Then 

f{x)  —fix,)  =  {s„{x)  -  .s„(.r„)  }  +  r,^{x)  —  r,(a-o). 

Wfi  will   show   that   the    absolute  value   of   eacii   of  the  (quantities 
l'\(-'')  — '\(-'^*o)|,   rjx),    7;(.ro)   is    less    than    \e.    if    8   is   properly 
chosen  and    |  x  —  a-„  |  <:^  8.     Fioiii  this   follows   that  the   absolute 
value  of  f(x)  — fi-^'o)  i^  less  than  e;   hence  the  proposition. 
Let  "the  remainder  in  the  JVf-series  be  denoted  by  R^  : 

and  let  n  be  so  chosen  that  Ix  <'  At,  and  then  held  fast.  Then, 
since 

l«„(^')  I    <    ^^„, 

•  •  ■  •  • 

it  follows  that 

I  '•,.(•'•)  I    <    K 

*  The  absolute  value  of  a  quantity  .1  shall  from  now  on  be  denoted  by    |  -4  |  . 


54         CONTINUIXr,    INTEGRATION,    DIFFEREXTIATION.        §§37,38. 

for  all  values  of  x  at  once,*  or 

I  ';(-^-)  I  <3e,  a<a;</3. 

Since  s^(x)  is  the  sum  of  a  fixed  number  of  continuous  functions, 
it  is  a,  continuous  function  and  hence  S  can  be  so  chosen  that 

I  \  (^)  —  K  (-'"o)  I  <  ^^  ,  I  a;  —  a-o  I  <  8  . 

Hence 

|/(a-)_/(;r,)  I  <e,  I  a5_a-o  |<S, 

and  tlie  theorem  is  proved. 

Exercise.     Sliow  that  the  series 

sin  a;        sin  8  a-        sin5.T 

— —    •       •       •       •       • 

12  3^      ^      5-^ 

converges  and  represents  a  continuous  function. 

38.  The  general  test  for  continuity'  just  obtained  can  be  applied 
at  once  to  power  series 

Theorem  2.  A  jjoiver  series  represents  a  continuwxs  function 
vjithin  its  interval  of  convergence.  The  function  may  however  he- 
come  discontinuous  on.  the  boundary  of  the  interval. 

Let  the  series  be 

f(x)  =  Go  -f  a^x  -f-  a.2X~  -|- , 

convergent  when  —  r  <^  x  <^  r ;  and  let  (a,  ^)  be  any  interval  con- 
tained in  the  interval  of  convergence,  neither  extremity  coinciding 
with  an  extremity  of  that  interval.  Let  X  be  chosen  greater  than 
either  of  the  quantities    |  a  |  ,    |  /3  |  ,  but  less  than  r.     Then 

I  ««^«"  I  <  I  ««  I  ^"^  a<x<(3; 

and  the  series 

I  a,  I  +  I  a,  I  X  +  I  a,  I  X^  + 

converges.     Hence  if  we  set 

M  =  \  a    I  X« ,  , 

the  conditions  of  Theorem  1  will  he  satisfied  and  therefore  f(x)  is 
continuous  throughout  the  interval  (a,  13). 

By  the  aid  of  this  theorem  the  following  theorem  can  be  readily 
proved. 

*  It  is  just  at  this  point  that  the  restriction  on  a  convergent  series  of  con- 
tinuous functions,  whicli  the  theorem  imposes,  comes  into  play.  Without  this 
restriction  this  proof  would  be  impossible  and  in  fact,  as  has  already  been 
pointed  out,  the  theorem  is  not  always  true. 


§  38.  CONTINUITY,    INTEGUATION,    DIFFEUENTIATION.  55 

Theorem.  If  a  j)oioer  series  vanishes  for  all  values  of  x  lying  in 
a  certain  interval  about  the  j)oint  cc  :=  0  : 

0  =  «„  "h  ^'i-^'  "h  ''2  -''^  ~h 5  —  I  <^x  <^l , 

then  each  coefficient  vanishes  : 

Oo  ^0,     tti  ^  0,      

First  put  X  :=  0 ;  theu  fto  =  0  aud  the  above  equation  can  be 
written  iu  the  form 

0  =:  X  (tti  -|-  ttoX  -{- ) 

From  this  equation  it  follows  that 

0  :=  ai  -\-  a^x  -{- 

provided  x  =h  0 ;  but  it  does  not  follow  that  this  last  equation  is 
satisfied  when  x  =  0,  and  therefore  tt^  cannot  be  shown  to  vanish  by 
putting  X  =  0  here  as  in  the  previous  case.  Theorem  2  furnishes  a 
conveaioiit  means  of  meeting  this  dilliculty.     Let 

/i(a-)  =  ai  4-  a,.»-  + 

Then  since /i(«)  is  by  that  theorem  a  continuous  function  of  x 

lim  ./;  (x)  =  f  (0)  =  tti . 
x  =  0 

But  lim    /;  (.r)  =  0  ;  .-.  a^  =  0  . 

x  =  0 

By  repeating  this  reasoning  each  of  the  subsequent  coefficients  can 
be  shown  to  be  0,  and  thus  the  theorem  is  established. 

CoROLLAUY.  //'  tico  poirev  sevies  hare  (he  same  value  for  all 
values  of  X  iu  a»  interval  about  (he  point  x  =.  0,  (heir  coefficients 
<(re  respectively  equal: 

ao-\-<iiX-\-a.,x--\- =  ba-\-biX-{-b.X'-{- ;  —  ?<a.'</, 

Oo  =:  6o?  ^'i  ^=  ^hi  6tC. 

Transpose  one  series  to  the  other  side  of  the  equation  and  the 
proof  is  at  once  obvious. 

The  Determinadon  of  the  Coefficients  c.  It  was  shown  in  §  36 
that  the  quotient  of  two  power  series  can  be  represented  as  a 
power  series. 

,^,  +  /,,+  ,,..+ ^,^^,^,^,^,.^ 

f'oH-  «i-'--|-  'l'■2•^•^+ 

By  the  aid  of  tlu'  theorems  of  this  paragraph  a  more  convenient 
mode  of  determining  the  coefficients  c  can  be  established.  Multiply 
each  side  of  the  equation  by  the  denominator  series  : 


56 


CONTINUITY,    INTEGRATION,    DIFFERENTIATION. 


§38. 


bQ-\-biX-\-b2X^-\-  •  •  '=:(aQ-\-aiX-\-a2X^-\-  •  •  ')(co-\- CiX-\- CoX^-f^-  •  •  •) 
=  tto  Co  -f  (tti  Co  +  ao  ci)  X  -\-  (a^  Co  +  «!  Ci  +  ^o  ^2)  a^^  + 


Hence 


C>o CIqCo 


61  =   OiCo  +  CloCi 

60  =  Oo-^o  H-  «'iCi  +  aoC2 


A  simple  mode  of  solving  these  equations  for  the  successive  c's  is 
furnished  by  the  rule  of  elementary  algebra  for  dividing  one  poly- 
nomial by  another, 

Qiiotient:     [  q^^  _|_  g^  x  -\-  C^X^ -\- 


&0 


x-\-'bo 

«._,  Co 


x''-{-h, 


x^-\- 


^0  H"  «i  ^  H-  «2  ^^  H"  ^z  ^^^  -\- 


(pQ  —  «o  Co)  +  {bi  —  tti  Co) 

«oCi 


^+(^2  — 02^0) 


a'''+(&3  — «3fo) 


iC^-f- 


(&1  —  aiCo  —  aoCi).T -|-  (62  —  02'-'o  —  «iCi) 


^'0^2 


'^''  H-  (&3  —  «3('o  —  «2Ci)  a;^  H- 

ttjC2 


{h.  —  a.,r^  —  a^(\  —  a^,c^)x^-\-{b^  —  a^c^  —  a^c^  —  a^c.^x^-\- 

etc. 

The  equations  determining  the  c's  are  precisely  the  condition  that 
the  first  term  in  the  remainder  shall  vanish  each  time. 

For  example,  to  develop  tan  .t,  divide  the  series  for  sin  x  by  the 
series  for  cos  x. 

Quotient:       \^   ^  -\-  ^  X^  -\-  ^^  x"-  -\- 

^  "(T  •*-"       I     T2'T7  '^       r 


i  ^  *^      ~\       TTX  **^ 


^X  3  0     '^        I 

^^-    i    x^^ 


ete. 
Hence 

tan  X'  ^  a;  -(-  ;^  .c^  ~h  t^'  ^^  ~f~ 

This  method  is  applicable  even  to  the  case  treated  in  the  corol- 
lary, §  36. 

Exercise.     Develop 

1  \2  —  bx-\-  x^ 


1  +  a;' 


3  +  .r  -(-  a;' 


.7     ' 


csCa; . 


§39, 


CONTINUITY,    INTEGRATION,    DIFFERENTIATION. 


57 


39.  The  Integration  of  Series  Term  by  Term.  Let  the  eontinuou3 
function /(x)  be  represented  by  an  infinite  series  of  continuous  func- 
tions convergent  througliout  the  interval  (a,  /?)  : 

f{x)  =  n,  {x)  +  n,  (X)  -h ,  a  <  X  <  ^  .      {A) 

The  problem  is  this  :  to  determine  when  the  integral  of  f{x)  will  be 
given  by  the  series  of  the  integrals  of  the  terms  on  the  right  of 
equation  (^1) ;  i.  e.  to  deterniinc  wlioii 

f^f(x)dx=    rn,{x)clx-{-    l\i,{x)dx-\- {B) 

will  be  a  true  equation.  The  right  hand  member  of  {B)  is  called 
the  term  by  term  integral  of  the  w-series. 

Let 

s,{x)  =  v^ix)  +  «iO«)  -f +  «„-i(a;), 

f{x)  =  s„(x)^r„(x). 
Then 

f^f{x)dx=    f^s,Xx)dx^    rr„{x)d 

t/a  t/a  tJa 

or 

Jf(x)dx=z  I     UQ(x)dx-\-  j     rii(x)dx-\- • ' 

+  /     r,Xx)dx. 

Hence  tlie  necessary  and  sufficient  condition  that  (B)  is  a  true  equation 
is  that  hm       f^r(x)dx=0. 

To  obtain  a  test  for  determining  when  this  condition  is  satisfied, 

plot  the  curve 

y  =  r„  (x)  . 


X 


+ 


X'""-' 


(x)dx 


y 

OL 

» 

y-  r.(X) 

V  —  Z'n 

Fig.  9. 


The  area  under  this  curve  will  represent  geometrically 


i 


r„(-v)dx. 


58  CONTINUITY,    INTEGRATION,    DIFFERENTIATION.  §  39. 

Draw  lines  through  the  highest  and  lowest  points  of  the  curve  parallel 
to  the  x'-axis.  The  distance  p„  of  the  more  remote  of  these  lines 
from  the  .T-axis  is  the  maximum  value  that  |  /•„  (x)  |  attains  in  the 
interval.  Lay  off  a  belt  bounded  by  the  lines  y  =  p,,  and  y  =:  —  p„. 
Then  the  curve  lies  wholly  within  this  belt  and  the  absolute  value  of 
the  area  under  the  curve  cannot  exceed  the  area  of  the  rectangle 
bounded  by  the  line  ?/  =  p„ ,  or  (/?  —  a)  p„ .  This  area  will  converge 
toward  0  as  its  limit  if  * 

lim   p„=  0, 

and  thus  we  shall  have  a  sufficient  condition  for  the  truth  of  equation 
(B)  if  we  establish  a  sufficient  condition  that  the  maximum  value  p„ 
of  I  ?•„  (x)  I  in  the  interval  (a,  /3)  approaches  0  when  n  =i  cc  .  Now 
we  saw  in  the  proof  of  Theorem  1  that  if  the  series  (A)  satisfies  the 
conditions  of  that  theorem, 

Hence  any  such  series  can  be  integrated  term  by  term  and  we  have 
in  this  result  a  test  sufficiently  general  for  most  of  the  cases  that 
arise  in  ordinary  practice.     Let  the  test  be  formulated  as  follows. 

Theorem  3.     Series  (A)  can  alivays  be  integrated  term  by  term,  i.  e. 

Jf(x)dx=:  I      UQ(x)dx-\-  I      Ui(x)dx-\- 

will  be  a  true  equation,  if  a  set  of  positive  numbers  Mq,  Mi,  M^, , 

independent  of  x,  can  be  found  such  that 

1)  I  >^:X^)  I    <    -^4,       a<x<l3,       n  =  0,1,2, ; 

2)  Jfo  +  ^/i  +  3^2  + 

is  a  convergent  series. 

The  form  in  which  the  test  has  been  deduced  is  restricted  to  real 
functions  of  a  real  variable.  But  the  theorem  itself  is  equally  appU- 
cable  to  complex  variables  and  functions.  It  is  desirable  therefore 
to  give  a  proof  that  applies  at  once  to  both  cases. 

*  This  condition  is  not  satisfied  by  all  series  that  are  subject  merely  to  the 
restrictions  hitherto  imposed  on  (A).  Not  every  series  of  this  sort  can  be 
integrated  term  by  term.  See  an  article  by  the  author  :  A  Geometrical  Method 
for  the  Treatment  of  Uniform  Convergence  and  Certain  Double  Limits,  Bul- 
letin of  the  Amer.  Math.  Soc,  2d  ser.,  vol.  iii,  Nov.  1896,  where  examples  of 
series  that  cannot  be  integrated  term  by  term  are  given  and  the  nature  of  such 
series  is  discussed  by  the  aid  of  graphical  methods. 


§§  39,  40.       CONTIXriTV,    I\TE(;UATIf^)V,    I)IFFEREXTFATK)\.         ^)d 

Keeping  tlie  notation  nsecl  above,  the  relation 

Jf(x)dx  =  I     iio (x) dx -\-  I     ai(x)dx-{- -f-  /     u„_i{x)dx 

+    /     r„(x)dx 

still  holds  and  the  proof  of  the  theorem  turns  on  showing  that  the 
hypotheses  are  sutticient  to  enable  us  to  infer  tliat 

1""      rrAx)dx=0. 

Let  the  remainder  of  the  3f-series  be  denoted  as  in  §  37  by  i2„ : 

i2,.=  3/„  +  3/„^,+ 

Then  it  follows,  as  in  that  paragraph,  that 

I  r„  (x)  I    <    7?,. . 
Now 

I      f^r,Xx)dx         <     P  I  r„(x)  \  ■  \  dx  \    <    RJ, 

I     tJa  tJa 

the  second  integral  being  extended  along  the  same  path  as  the  first, 
and  I  denoting  the  length  of  the  path,     liut    lim    (BJ)  =  0;   hence 

I      r,^(x)dx  converges    toward   0  when  n  =:  x    and   the   proof   is 

complete. 

40.  We  proceed  now  to  apply  the  above  test  to  the  integration 
of  some  of  the  more  common  forms  of  series. 

First  Application :  Poioer  Series.  A  poioer  series  can  be  integrated 
term  by  term  throughout  any  interval  (a,  (3)  contained  in  the  interval 
of  convergence  and  not  reaching  out  to  the  extremities  of  this  interval: 

|a|<r,  1/3  I    <r. 

Let  the  series  be  written  in  the  form 

f(x)  =  a,,  -f-  a,.r  -|-  a.,.r-  -(- 

and  let  X  be  chosen  greater  than  the  greater  of  the  two  quantities 

I  a  I  ,    I  /?  I  ,  but  less  than  r.     Then 

I  a„x"  I    <    I  a,.  I  X\  a<x<P, 

and  if  we  set  |  a„  |  X"  =  .V„ , 

the  conditions  of  the  test  will  be  satisfied. 


60  CONTINUITY,    INTEGRATION,    DIFFERENTIATION.  §  40. 


In  particular 

Jo  •''"'^^  '    "^2    ■    -3 


f(x)dx  =  aoh -\-  ai  — -{-  a^  ^ -\- ,      |  /i  |  <  r 

0 


If  when  X  =  r  or  —  r,  the  series  for  f(x)  converges  absolutely, 
then  h  may  be  taken  equal  to  r  or  —  r.  If  however  the  series  for 
f(x)  does  not  converge  absolutely  or  diverges  when  a;  =  r  or  —  r,  it 
may  nevertheless  happen  that  the  integral  series  converges  when 
h  z=  r  or  —  r.  In  this  case  the  value  of  the  integral  series  will  still 
be  the  integral  of  f(x).     Thus  the  series 

diverges  when  x  z=  I ;  but  the  equation 


-o' 


x 


/'     dx     _  Ji^       h^  _ 

0  r+ii -''~j^j~ 


still  holds  when  h  =:  1  : 

log2=  i_^  +  ^_^+---.- 
The  proof  of  this  theorem  will  be  omitted. 

Second  Application:  Series  of  Powers  of  a  Function.  Let  <f>(x)  he 
a  continuous  function  of  x  ayid  let  its  maximum  and  minimum  values 
lie  between  —  r  and  r  ivhen  a  '^  x  \  fi.     Let  the  power  series 

converge  lolien  —  r  <^  y  <^r.      Then  the  series 

f{x)=  ao-\- ai<f>(x)-{- ao[cl>{x)Y^ 

can  be  integrated  term  by  term  from  a  to  /3 ; 

Jf{x)dx  =  ao  I     dx-\-ai  j     (f>(x)dx-\-a2  I     [cl>(x)Ydi 

For  if  Y  be  so  taken  that  it  is  greater  than  the  numerically  greatest 
value  of  <^  (cc)  in  the  interval  a  ^  a-  ^  /3,  but  less  than  r,  then 

1)  I  «„|    I  <^(.i-)  I  »    <    |a„  I  Y\ 

2)  I  tto  \+\ch\y+\ch\Y'+ 

converges ;   and  if  we  set 

I  a„  I   r«  =  M,, , 

the  conditions  of  the  test  will  be  satisfied. 

Thus  the  integrations  of  §§  24,  25  are  justified. 


Lt  + 


§§  40,  41.       CONTINUITy,    IXTEGKATION,    DII  TEUENTIATIOX.         ()1 

Third  Application.     If  the  function  <f>(x)  and  the  series 

«o  +  "uV  +  "2.'/^  + 

satisfy  the  same  conditions  as  in  the  precedinrj  theorem  and  if  ^{/(x)  is 
any  contimious  function  ofx,  then  the  series 

f(x)  =  a^il/(x)  -\-  a,xp(x)  <t>(x)  +  a.,ip(x)  \_<f>(x)Y  -\- 

can  he  inter/rated  term  by  term. 

The  method  of  proof  has  been  so  fully  illiistratcd  in  the  two  pre- 
ceding iippncations  that  the  detailed  coiistruetioii  of  the  proof  may 
be  left  as  an  exercise  to  the  student. 

This  theorem  is  needed  in  the  deduction  of  Taylor's  Theorem  from 
Cauehy's  Integral. 

Examples.     1.  Compute 

TT 

/x^e'^'dx,                 I      yj  smx  dx. 
2.  Show  that 
l£cos(xsiu<t^d.t.=  l-^+^^^-^^-^ 


Hitherto  the  limits  of  integration  have  always  been  the  limits  of 
the  interval  considered,  a  and  (3.  It  becomes  evident  on  a  little 
reflection  that  if  any  other  limits  of  integration,  X(„  x,  lying  within 
the  interval  (a,  ft)  are  taken,  Theorem  0  will  still  hold  : 

Jf*x                               /*x                                  /*x 
f(x)dx=   I     UQ(x)dx  -\-  I     ni(x)dx  -\- 

a  ^  .I'o  ^  /?,  a  ^  a:  ^  /3. 

For,  all  the  conditions  of  the  test  will  hold  for  the  interval  (xq,  x)  if 
they  hold  for  the  intei-val  (a,  /?). 

41.  The  Differentiation  of  Series  Term  by  Term.  Let  the  function 
f(x)  he  represented  by  the  series: 

f(.v)=  u,(.r)-\-u,(x)-\- 

throughout  the  interval  (a,  /3).      Then  the  derivative  f  (x)  will  he  given 
at  any  point  of  the  interval  by  the  series  of  the  derivatives : 

f(x)=u',(x)-\-n\  (..■)-}- 

provided  the  series  of  the  derivatives 

w'o(.r)-h^/\(.r)-h 

satisjies  the  conditions  of  Theorem  1  throughout  the  interval  (a,  /3). 


62  CONTINUITY,    INTEGRATION,    DIFFERENTIATION.  §  41. 

Let  the  latter  series  be  denoted  l)y  cf>  (^x)  : 

4>  (x)  —  u'o  (x)  -\-  u\  (x)  + 

We  wish  to  prove  that 

ct.(x)=f'(x). 

By  Theorem  1  the  function  <^  (a.*)  represented  by  the  w'-series  is  con- 
tinuous and  by  Theorem  3  the  series  can  be  integrated  term  by  term  : 

(J3(x)dx:=   I     « 'o (a;) d X- -f-   /     v'i(;x)dx  -\- 

a.  fj  a  *J  a. 

=   {"oO«)—  "u(a)}    +    {»i(-^')—  «i(«))   + 

=  f(x)-f(a). 
Hence,  differentiating, 

<t>(x)  =z  /'(x),  q.  e.  d. 

Exercise.     Show  that  the  series 

cos  X        cos  3  X        cos  5  x 

can  be  differentiated  term  by  term. 

By  the  aid  of  this  general  theorem  we  can  at  once  prove  the  follow- 
ing theorem. 

Theorem.  A  poiver  series  can  be  differentiated  term  by  term  at 
any  point  within  (but  not  necessarily  at  a  point  on  the  boundary  of) 
its  interval  of  convergence. 

Let  the  power  series  be 

convergent  when    \  x  \  <^  r,  and  form  the  series  of  the  derivatives  : 

«!  -|-  -la^x  -f-  303.1-2  -f- 

Then  we  want  to  prove  that  if    |  Xq  |    <^  ?•, 

/'(a-o)  =  «!  -\-  2a2.ro  -h  3a3.V  -f 

It  will  be  sufficient  to  show  that  the  series  of  the  derivatives  con- 
verges when  I  .X'  I  <^  ?• ;  for  in  that  case,  if  X  be  so  chosen  that 
I  iCo  I  <C  ^  <C  ^'1  the  conditions  of  the  test  will  be  fulfilled  through- 
out the  interval  ( —  X,  X).  We  can  prove  this  as  follows.  Let  x' 
be  any  value  of  x  within  the  interval  ( —  ?•,  r)  :  —  r  <^  a;'  <^  r,  and 
let  X'  be  so  chosen  that    |  ic'  |  <^  X'  <^  r.     The  series 

jao  I  +  1%  I  X'+  la^  I  X'^+ 

converges.     It  will  serve  as  a  test-series  for  the  convergence  of 

I  a,  l-f  2  I  a^  I    I  X'  I  +  3  I  a,  I    |  a;'  |  ^ 


§  41.  CONTINUITY,    INTEOKATION,    DIFFERENTIATION.  63 

if  it  can  be  shown  tliat 

n  I  a„  I    !  X'  j  »-»    <    I  a„  |  X'" 
from  some  definite  point,  n  =  m,  on.     This  will  be  the  case  if 


x' 


n 


X 


ri 


<    I  a;'  I  ,  nym 


But  the  expression  on  the  left  approaches  0  when  ?i  =  x  ,  for 
\  x'  \  I  X'  is  independent  of  n  and  less  than  1  ;  the  limit  can 
therefore  be  obtained  by  the  usual  method  for  evaluating  the  limit 
00  •  0.  Hence  the  condition  that  the  former  series  may  serve  as 
test-series  is  fulfilled  and  the  proof  is  complete. 

Exercise.     From  the  formula 
1 


1  +  X  +  a;2  -f  a-3  + 


1  —X 

obtain  by  differentiation  the  developments  for 

1  1  1 


(1  —x)^'       (1  —  a-)*'  (1  —.';)"' 

and  show  that  they  agree  with  the  corresponding  developments  given 
by  the  binomial  theorem. 


APPENDIX. 


A    FUNDAMENTAL    THEOREM    REGARDING    THE    EXISTENCE 

OF   A    LIMIT. 

It  was  shown  in  §  13  that  if  the  variable  s^  approaches  a  limit 
when  n  increases  indefinitely,  then 

approaches  the  limit  0  when  p  is  constant  or  varies  in  any  wise  with 
n.  And  it  was  stated  that  if,  conversely,  the  limit  of  this  expression 
is  0  when  n  zn  go  ,  no  matte}'  how  we  allovo  p  to  vary  loith  n,  then  s^ 
will  ai)proach  a  limit.  This  latter  theorem  is  important  in  the  theory 
of  infinite  series.  It  is  however  only  a  special  case  of  a  theorem 
regarding  the  existence  of  a  limit,  which  is  of  fundamental  impor- 
tance in  higher  analysis. 

Theorem.     Let  f(x)  he  any  function  of  x  such  that 

Urn  lf{x')  —/(a;")]  =  0 

when  x'  and  x",  regarded  as  independent  variables,  both  become  infi- 
nite.     Then  f(x)  approaches  a  limit  when  x  =1  x  . 

We  will  begin  by  stating  precisely  what  we  mean  by  saying  that 
f(x')  —  /O^")  approaches  the  limit  0  when  x'  and  x",  regarded  as 
independent  variables,  both  become  infinite.  AVe  mean  that  if  X  is 
taken  as  an  independent  varialile  that  is  allowed  to  increase  without 
limit  and  then,  corresponding  to  any  given  value  of  X,  the  pair  of 
values  (x',  x")  is  chosen  arbitrarily  subject  only  to  the  condition  that 
both  x'  and  x"  are  greater  than  X  (or  at  least  as  great),  the  quantity 
f(x')  — /(*")  will  then  converge  towards  0  as  its  limit.  In  other 
words,  let  e  denote  an  arbitrarily  small  positive  quantity.  Then  X 
can  be  so  chosen  that  * 

I  f(x')  —f{x")  I  <  £  ,         if         x'>  X        and  x"  >  X. 

We  proceed  now  to  the  proof.  Let  us  choose  for  the  successive 
values  that  c  is  to  take  on  any  set  Cj,  cg*,  £3, steadily  de- 
creasing and   approaching  the  limit  0  ;  —  for   example  the  values 

1,  ^,  ^, )  «,  =  1/i-     Denote  the  corresponding  values  of 

X  by  Xi,  X2,  X3 Then  in  general  these  latter  values  will 

*  For  the  notation  cf.  foot-note,  p.  53. 


APPENDIX.  65 

steadily  increase,  and  we  can  in  any  case  choose  tliem  so  that  they 
do  always  increase. 

Begin  by  putting  e  =  q  : 

I  f{x')  -/(.V)  |<  c, ,         .'.'  >  X\  ,         X"  >  X,. 

Assign  to  a;'  the  value  X^.     Then 

\f(X,)-fix)  |<ci, 

i.e.  '  /(XO-q</(a;)</(XO  +  ci 

for  all  values  of  x  greater  than  X^.  The  meaning  of  this  last  rela- 
tion can  be  illustrated  graphically  as  follows.  Plot  the  point  f(Xi) 
on  a  line  and  mark  the  points  /(X^)  —  cj  and  f(Xi)  -f-  tj.  Then 
the  inequalities  assert  that  the  point  which  represents  f(x)  always 
lies    within    this    interval,  whose   length  is    2ei,  pro\ided  x^X^. 


y<v-  *.  /^'  f(^i> 


^-t. 


I 


4—1 


=  1*1!.  :  ',     =:Pi 


A 


Fio.  lu. 

Denote  the  left  hand  boundary  f{Xi)  —  e^  of  this  interval  by  a^, 
the  right  hand  l)Oundary  f(Xi)  -|-  c^  by  /Sj.  Then,  to  restate  con- 
cisely the  foregoing  results, 

«!  <  f(.^)  <  /3i         if         ^  >  ^i ;  A  —  ai  =  2  €,. 

Now  repeat  this  step,  choosing  lor  c  the  value  co ' 

|/(X,)— /(.1-)    |<e.,, 
i.  e.  f{X,)  —  c,  <  J\x)  <  f(X,)  +  c, , 

where  x  denotes  any  value  of  the  variable  x  greater  than  X.,.  Plot 
the  point  f(Xo) ;  this  point  Ues  in  the  interval  (ai,  fii).  Mark  the 
points  f(X^)  —  £2  and  f(Xo)  -\-  co .     Then  three  cases  can  arise  : 

(a)  both  of  these  points  lie  in  the  interval  (aj,  (3i) ;  let  them  be 
denoted  respectively  by  ao ,  ^2 ; 

(b)  /(-X'2)  lies  so  near  to  ai  that/(Xo)  —  co  f^^ll^  outside  the  inter- 
val ;  in  this  case,  let  a^  be  taken  coincident  with  a^ :  ao  =  a^  ;  the 
other  point /(Xo)  -\-  e-2  will  lie  in  the  interval  (ai,  jSi)  and  shall  be 
denoted  by  ySo  5 

(c)  /(X2)  lies  so  near  to  /3i  that  f(X..)  -\-  e.,  falls  outside  the  inter- 
val ;  in  this  case,  let  ySj  be  taken  coincident  with  (3i:  /Sa  ==  i^i ;  the 
other  point  f(Xo)  —  €0  will  lie  in  the  interval  (aj,  /?i)  and  shall  be 
denoted  bv  a... 


6G  APPENDIX. 

In  each  one  of  these  three  cases 

ao  <  f{^)  <  A         if         a;  >  X2 ;  /?2  —  a2  <  2  e^ . 

The  remaiudev  of  the  proof  is  extremely  simple.  The  step  just 
described  at  length  can  be  repeated  again  and  again,  and  we  shall 
have  as  the  result  in  the  general  case  the  following : 

Now  consider  the  set  of  points  that  represent  a^,  ug,  .  .  .  a,,  .  .  . 
They  advance  in  general  toward  the  right  as  i  increases, — they 
never  recede  toward  the  left, — but  no  one  of  them  ever  advances 
so  far  to  the  right  as  /Jj.  Hence,  by  the  principle*  of  §  4,  they 
approach  a  limit  A.  Similar  reasoning  shows  that  the  points  repre- 
senting /3i,  /^a,   .   .   •  A,   •   •   •  approach  a  limit  B.     And  since 

these  limits  must  be  equal :  A  =  B. 

From  this  it  follows  that  f{x)  converges  toward  the  same  limit. 
For 

and  if  when  x  increases  indefinitely,  we  allow  i  to  increase  indefi- 
nitely at  the  same  time,  but  not  1-0  rapidly  as  to  invalidate  these  in- 
equalities, we  see  that /(a;)  is  shut  in  between  two  variables,  a,  and  ;8,-, 
each  of  which  approaches  the  same  limit.  Hence /(a;)  approaches 
that  limit  also,  and  the  theorem  is  proved. 

In  the  theorem  in  infinite  series  above  quoted  n  is  the  independent 
variable  x,  s^  the  function  f(x)  ;  the  expression  s^^^  —  s^  corre- 
sponds to  f{x')  — /(a;");  and  thus  that  theorem  is  seen  to  be  a 
special  case  of  the  theorem  just  proved.  The  domain  of  values  for 
the  variable  x  is  in  this  case  the  positive  integers,  1,  2,  3, 

Another  application  of  the  present  theorem  is  to  the  convergence 
of  a  definite  integral  when  the  upper  limit  becomes  infinite.     Let 

f(x)=  rct>{x)dx. 

tJ  a 

4>  (x)  d  X—  I      </)  {x)  dx  =  I     cl>  (x)  d  X. 

J'' a-/ 
(^  {x)  dx  =z  0 

*  This  principle  was  stated,  to  be  sure,  in  the  form  5',/ I>  5  if  ?;'>  w; 
but  it  obviously  continues  to  hold  if  we  assume  merely  that  S,,'  ^  S,,  when 
n'  >  n. 


APPENDIX.  67 

when  x'  and  a;",  regarded  as  independent  variables,  both  become  in- 
finite, tlie  integral 


f 


(ji  (x)  dx 


is  convergent.  The  domain  of  valnes  for  the  variable  x  is  in  this 
case  all  the  real  quantities  greater  than  a. 

In  the  foregoing  theorem  it  has  been  assumed  that  the  independent 
variable  x  increases  without  limit.  Tiie  theorem  can  however  be 
readily  extended  to  the  case  that  x  decreases  algebraically  indefi- 
nitely or  approaches  a  limit  a  from  either  side  or  from  both  sides. 
In  the  tirst  case,  let 

x  =  —y; 
in  the  second,  let 

,    1 

X  =z  a  -\ — 

y 

if  X  is  always  greater  than  its  limit  a ;  let 

1 
X  =^  a 

y 

if  X  is  always  less  than  a.     Then  if  we  set 

f(x)=  c/>(.v) 

and  the  function  ^  (//)  satisfies  the  conditions  of  the  theorem  when 
y  =z  -\-  cc  ,  <f>  (y)  and  hence /(.r)  will  approach  a  limit.  Finally,  if  x 
in  approaching  a  assumes  values  sometimes  greater  than  a  and  some- 
times less,  we  may  restrict  x  first  to  approaching  a  from  above, 
secondly  from  l)elow.  In  each  of  these  cases  it  has  just  been  seen 
that /(a;)  approaches  a  limit,  and  since 

lim[/(.i•')-/(.^•")]  z=0 

where  x'  and  x"  may  now  be  taken  the  one  above,  the  other  below  a, 
these  two  limits  must  be  equal.  We  are  thus  led  to  the  following 
more  general  form  of  statement  of  the  theorem. 

Theorem.     Let  f(x)  he  such  a  function  ofx  that 

Urn  [/(x')— /(x")]  =  0 

when  x'  and  x",  regarded  as  independent  variables,  approach  the  limit 
a  from  above  or  from  below  or  from  both  sides,  or  become  jwsitively  or 
negatively  infinite.  Then  f(x)  apjrroaches  a  limit  when  x  approaches 
the  limit  a  from  above  or  from  below  or  from  both  sides,  or  becomes 
positively  or  negatively  infinite. 


68  APPENDIX. 


A   TABLE   OF   THE   MORE   IMPORTANT   FORMULAS. 

The  heavy  line  indicates  the  region  of  convergence. 

._J_  =  1  +  a;  +  x-^  +  ..3  + 

—  1  0  1 


a  —  bx       a       cr  cr  cr 


■r  II  r 


r  =  -  numerically. 


x^    ,     x^         X* 


log  (1+X)=.T  --  +  ---  + 


—  1  0  1 


(^  +  f  +  f+ ) 


log  [^=2(0.  +  :^+:^  + 


—  10  1 


(l  +  x)M=l+^.  +  tl^..-^  +  ?iOi=i|i^.^  + 


—  1 


^  =  1  _  2ic  +  3.^•2  —  ix^  -\- 


{l-\-x) 

—  1 


APPENDIX.  69 


I     O  '     ■>  .    1  I     •>  .  /I   .  c  I 


V  1  —  a;2  '2         '21         '    2  •  4  •  6 

—  1  0  1 


Vl_a-»^l-^x»-^j^-^.. 


—  1  (I  1 


/V.2  ,,.8  ,.4 

«^=  1+^  +  ^  +  3.  +  .!,+ 


a;*    ,     x^         x'' 


sinx  =  x---^^--^ 


II 


a;2     ,     a;*         .j;« 


cosa.'=  i__-f  + 


4!        6! 

0 


tan  a;  m  a;  H-  ^  .r^  -f-  j-^  x^  -|- 


-  ^  0  •■> 


cot  ar  — ^  .)•  —    \  x^  -\- 

X 


■  IT  < '  E 


70  APPENDIX. 


sec x=l-\-^x^-\-^x*-\-  '   • 

—  TT  0  Y 


.      ,  ,    1  a;3        1  .  3  a;5    ,    1  •  3  •  5  a;''    , 


—  1  0  1 


x^    ,    x^        .v'' 


tan   ^x  r=  X 1 '■ — 1- 

3  0  i 


—  1  0  1 


f(Xo-\-?l)  =  /(Xo)-\-f'(x,)h-\-r(Xo)^^  H- +/(«)(a:„  +  ^/,)  ^" 


0<^<1 


f{x,  -j-h)=  /(.i-o)  +  hf  (a-o  4-  ^/i) 
Jo    ^/  1  — k'^sin^<f> 


TT 

E 


=    j     ^  I  —  k'^sm^ff,  d<f>  = 


Ari'ENDIX.  71 

For  small  values  of  x  the  following  equations  are  approximately 
correct. 

f(a  +  x)=f{a)-\-f'ia)x 

(1  -^  x)"'=  1  -(-  ma; 
(I  -\-  xy  =  \  -\-  -Ix 

VT  +  ^rrr   1  +  ^x 

=  \  —  2x 

ii-\-xy 

,    ^        =  1  — Ax 

V  1  +  cc 

If  X,  y,  z,  to, are  all  numerically  sn.all,  then,  approximately, 

(1  +  .T)  (1  +  2/)  (1  +  ^) =  1  4-  .0  +  y  +  ^  + 

sin  a;  ==  x         or         x  —  ^  x^ 

cos  X  rr:   1  Or  1   —  i  .i'^ 

tan  X'  z=  X-       or         .r  -f-  j^  x^ 
sin  (a  -|-  a*)  —  sin  a  =  x  cos  a 
cos  (a  -|-  x)  —  cos  a  =  —  x  sin  a 
log  (a  -\-  x)  —  log  a  r=  - 


! 


nr. '^nuTHfRfi 


RFGIONAL  LIBRARY  FACILITY 


nil 


AA    000  507  745    b 


